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JHEP03(2021)148 and d,e Springer , and at the March 15, 2021 : January 29, 2021 : November 13, 2020 : Published Accepted MoEDAL-MAPP Received Zeren Simon Wang c [email protected] detector, the future far-detector , and , , CMS / Published for SISSA by https://doi.org/10.1007/JHEP03(2021)148 MATHUSLA ATLAS , Julian Y. G¨unther, SMEFT Lagrangian. We perform simulations to FASER c ν . We study scenarios where sterile are pre- , [email protected] , SHiP [email protected] SMEFT). The production and decay of sterile neutrinos , ν CODEX-b , . 3 ANUBIS , Herbert K. Dreiner, 2010.07305 a The Authors. a,b AL3X c Phenomenological Models

We study the prospects of a displaced-vertex search of sterile neutrinos at , [email protected] Asia Pacific Center forHeadquarters Theoretical San Physics 31, (APCTP), Hyoja-dong, Nam-gu,E-mail: Pohang 790-784, South Korea [email protected] Upton, NY 11973-5000, U.S.A. Bethe Center for Theoretical PhysicsNußallee & 12, Physikalisches 53115 Institut Bonn, der Germany Department Universit¨atBonn, of Physics, NationalHsinchu Tsing 300, Hua Taiwan University, Amherst Center for FundamentalDepartment Interactions, of Physics, UniversityAmherst, of MA Massachusetts, 01003, U.S.A. RIKEN BNL Research Center, Brookhaven National Laboratory, b c e d a Open Access Article funded by SCOAP determine the potential reachoperators, of finding that high-luminosity these LHC experiments experiments are very in competitive probingKeywords: with the other searches. EFT ArXiv ePrint: We investigate the search sensitivitiesexperiments: for the proposed fixed-target experiment dominantly produced via rareand/or charm and dimension-six bottom operators in decays the through minimal mixing the Large ColliderModel Effective (LHC) Field in Theory thecan ( proceed framework via of the the standardthrough active-sterile -extended higher-dimensional neutrino operators Standard mixing arising intrinos the from are weak decoupled long-lived, current, their new as decay physics. well can as lead If to sterile displaced vertices neu- which can be reconstructed. Abstract: Guanghui Zhou field theory Jordy de Vries, Long-lived sterile neutrinos at the LHC in effective JHEP03(2021)148 3 8 12 32 8 9 6 – i – 3 14 8 17 19 9 21 23 18 30 31 25 27 28 12 19 19 22 20 17 19 1 34 16 6.2.7 ATLAS 6.2.2 MATHUSLA 6.2.3 ANUBIS 6.2.4 CODEX-b 6.2.5 MoEDAL-MAPP 6.2.6 AL3X 6.2.1 FASER 7.3 Flavor benchmark7.4 2 Flavor benchmark7.5 3 Flavor benchmark7.6 4 Flavor benchmark 5 6.3 Monte-Carlo simulation 7.1 The minimal7.2 scenario Flavor benchmark 1 6.1 SHiP 6.2 Experiments at the LHC 4.1 Sterile neutrino4.2 decays in minimal Sterile models neutrino decays from higher-dimensional operators 2.1 The effective2.2 neutrino Lagrangian Rotating to the neutrino mass basis 3.1 Sterile neutrino3.2 production in minimal Sterile models neutrino production from higher-dimensional operators 8 Discussion and comparison with9 other probes Conclusions 7 Numerical results 6 Collider and fixed-target analysis 4 Decay of sterile neutrinos 5 Theoretical scenarios 3 Production of sterile neutrinos Contents 1 Introduction 2 effective field theory extended by sterile neutrinos JHEP03(2021)148 43 44 44 45 ), is νββ 36 42 38 40 ]. Sterile neutrinos are thus a 40 39 10 0 V 42 M – 1 – s decaying into D 0 P ]. The sterile neutrino, also called heavy neutral D 5 0 P M – 1 M and → ) ) → s s ( ( 36 B D B ], while sterile neutrinos with a broad range of masses can account for the 9 – 6 While the observation of neutrino masses provides a hint for the existence of sterile In general, nothing forbids an additional Majorana mass term for the right-handed C.3 Form factors for C.4 The semi-leptonic decay of B.2 Definition of three-body decay form factors C.1 Form factors for C.2 Form factors for A.1 Charged currents A.2 Sterile neutrinoA.3 decays into neutral Sterile pseudoscalar neutrino mesons decays into neutral vector mesons B.1 Automizing three-body phase space integral calculations neutrinos, it does not specifyin their present-day mass and near scale. future They experiments. mightworks A very have large well number gone be of into experimental light the and andsterile theoretical accessible search neutrinos for only sterile interact neutrinos with in SM so-called fields minimal through scenarios, renormalizable where Yukawa interactions linked to explanations of otherdark problems of [ the SM. Lightbaryon sterile asymmetry neutrinos of can the accountwell-motivated Universe for solution through to [ acosmology. number of major outstanding issues in physics and neutrino field, leading toHowever, Majorana lepton mass number eigenstates can and be(BSM), lepton an such number (approximate) that violation symmetry low-energy (LNV). ofsuppressed. LNV extension signals, Sterile beyond neutrinos e.g. may the neutrinoless not SM only double account beta for decay neutrino masses, (0 but have also been forbids the generation ofmasses a of neutrino the mass viasterile other neutrino the elementary field Higgs to . mechanism, the(HNL), SM which is This [ generates a situation the right-handed gauge-singlet canand -1/2 be the field Higgs remedied and couples field byafter to through electroweak adding left-handed symmetry Yukawa neutrinos a interactions. breaking. This generates a Dirac neutrino mass 1 Introduction Neutrino oscillations have proven withoutStandard doubt that Model neutrinos of are particle massive particles. physics (SM) The contains no right-handed neutrino fields. This C Physical parameters, decay constants, and form factors B Sterile neutrino production in three-body decays A Two-body sterile neutrino production and decay processes JHEP03(2021)148 ] ], ], 23 29 18 – [ decay 21 decay in ], as well νββ 38 νββ , CODEX-b ], we present the 37 [ 7 28 [ ]. For sterile neutrino CMS ]/ 25 , 27 [ 24 . ], or models [ MoEDAL-MAPP 9 17 ], ATLAS 36 [ ], and discuss their potential in prob- we introduce the model framework of 41 2 – 39 ANUBIS [ ], ], Z’ models [ – 2 – 35 16 [ SHiP SMEFT corrections to sterile neutrino production ]. ν detailing the calculation of production cross sections 20 4 , AL3X 19 shows the theoretical scenarios considered by numerical ], goes through the different experiments we study in detail and 5 34 6 3 – 32 [ ] to be less than 10%. The latter become more important and 26 SMEFT) [ ν - threshold the primary production mode is through rare decays B ] for a review). Here we take a more general approach. In broad classes MATHUSLA ], grand unified theories [ ], 12 , 15 – 31 11 , . In a set of appendices, we present the details of the production and decay rate 13 30 [ 8 SMEFT operators. We calculate This paper is organized as follows. In section In this work, we study relatively light GeV-scale sterile neutrinos (see e.g. refs. [ ν SMEFT, followed by sections computations, as well aswe the employ. physical We parameters, conclude decay and constants, provide and an form outlook factor in input section study in this work, andand section briefly introduces thenumerical Monte-Carlo results simulation for procedure. both the Inresults minimal section are scenario compared and a with number asection of number flavor of benchmarks. other These experimental probes including 0 limits, showing that these experiments are complementary. ν and decay widths ofdimensional operators. the sterile Section neutrinos in both the minimal model and from higher- ing and decay processes andsearch perform sensitivities. simulations Our for simulationsdimension-six the show operators various that associated detectors, the to to experimentalmodel, BSM estimate adding reach scales their just is up one to strong, sterile a neutrino probing field, hundred we TeV. compare In our a results simple to existing 3 + 0 1 production channel fortheir future decays studies. lead to Ifconsider displaced a the broad vertices range sterile that ofFASER neutrinos (proposed) can LHC are be experiments: relatively reconstructed inas long-lived, the LHC proposed detectors. CERN SPS We experiment that are copiously producedmasses at below the the LHCof interaction mesons points with [ subleadinging contributions from MadGraph5 partonic 3.0.2 processes, [ even which dominant we for estimated heavier us- range sterile neutrinos. below about In this 5 GeV work and we choose hence to on focus the on rare the meson mass decays, and we leave the direct of local effectiveeffective operators field in theory ( the framework of the neutrino-extendedfor Standard LHC Model searchesduced for either somewhat via heavier direct neutrinos). production Such with sterile parton neutrinos collisions, can or be via pro- rare decays of mesons of BSM models, sterileenergies neutrinos through appear the sterile exchangemodels at of [ lower heavy energies, BSMwhich but contain fields. interact new fields at Examples that higher arethe are heavy details left-right compared of to symmetric these the models, electroweak scale. at low Independent energies of the sterile neutrinos can be described in terms (see refs. [ JHEP03(2021)148 ]. R ν 52 . – (2.1) (2.2) h matrix 50 2. , n / , where Λ ) 19 × 5 v . We define γ † ]). As lepton C   49 − is a 3 the CKM matrix. ν = SMEFT. = (1 SMEFT [ Y ) in general leads to V ν T ν C . 2.1 R,L . is the SM complex Higgs ν − P  is the charge conjugation . Y with c H . = C 1 , eq. ( and − mass + h R mass matrix. In general we can j, L C ¯ R decay and other LNV processes, R and M d ¯ n ν − M ij ν , T × ¯ V = Ψ ! ˜ νββ n HY C v h C = ¯ L 0 = i L + d c gauge group and interact with right-handed 1 + R ν R

R – 3 – 2 ¯ v M √ c R ¯ ν = , in terms of the projectors = 2 the lightest neutrino is massless [ 1 2 c is the lepton doublet, and  H n Ψ L − right-handed gauge-singlet neutrinos. ) can account for the observed active neutrino masses and R,L P . We work in the unitary gauge SM n ∗ 2.1 L = is a complex symmetric H 2 T = R 3, for which we have iτ , ¯ L M 2 L,R 2 (in case of , = , which satisfies the relation Ψ neutrinos, which are singlets under the SM gauge groups. The interac- 0 ˜ ]. The sterile neutrinos are assumed to be singlets under the SM gauge ≥ C γ H n 2 = 1 n 48 = i – iγ ) form the dimension-four and lower part of a more general Lagrangian con- c , ) − 42 i L 2.1 d = L,R 1 column vector of = 246 GeV is the Higgs vacuum expectation value of the Higgs real scalar C × v , as these are copiously produced and are sufficiently massive to produce GeV = (Ψ n denotes the SM Lagrangian, c In various popular extensions of the SM, right-handed neutrinos appear naturally, but The Lagrangian in eq. ( is the charge conjugate field of Ψ with Ψ c c L,R or SM as well as a settions of in eq. ( taining higher-dimensional operators that isWe often begin referred by to introducing asdenotes the the the scale higher-dimensional where operators we match at the microscopic a UV scale theory Λ to the are not completely sterile. For instance,nos in are left-right charged symmetric models under right-handed a neutri- gauge right-handed SU(2) and new scalarwell above fields. the If electroweak scale, such to bosonstrinos avoid exist, on experimental they the constraints. must other The besuggests right-handed hand, heavy, the neu- can with use of masses remain an light. EFT framework From where this the degrees point of of freedom are view, the the usual SM scale fields, separation mixing angles if number is explicitly violatedMajorana by neutrino the mass Majorana eigenstatesunless masses, additional and structure thus is to imposed 0 on the matrices except the Ψ matrix, Ψ where is an of Yukawa couplings. work in a basis where the charged and all are in their mass eigenstates, L doublet field with sterile neutrinos. This isleptogenesis an [ interesting mass rangegroup that and, appears at in thecoupling scenarios renormalizable to of level, the low-scale only lepton doublet. interact with The renormalizable SM part fields of via the a Lagrangian Higgs is given Yukawa by 2.1 The effectiveWe neutrino are interested Lagrangian in the productionwe and investigate decay the of sterile production neutrinos of at sterileb the neutrinos LHC. in In the particular, decay of mesons containing a single 2 Standard model effective field theory extended by sterile neutrinos JHEP03(2021)148 2 ≥ (2.3) (2.4) (2.5) (2.6) n , (6) T . For our ]. We focus C ). For 1 T 54 γ , ). The related 2.1 H.  † 20 s 2.1 π H α 4 ) R e  ) ) ν )) , µ R γ L in eq. ( = ¯ (5) R Qd R R ¯ ( Qν ν ν 4 R µ in eq. (  ( M (6) (6) T ) ¯ LdQν ψ  )(¯ )(¯ M ν ) ln c R u R C Y ¯ µ ¯ ν d dC Qu 1 8 and limit the set of effective ¯ Ld ¯ Lν ( ( − ¯ ( dγ , − 1 ( , = F ]. = C R (6) S (5) ν 57 (6) T , C L = 2 do not evolve under QCD at one loop. ] (6) duνe S (6) (6) LdQν LνQd (6) QuνL 56 γ T O 51 O O O Class 4  ]. This leaves us with the remaining six (6) duνe s , π ] involving one sterile neutrino field. 55 ,C α 4 O n ) 20  H Iµν ) , γ H – 4 – l µ = F H (6) LνQd H † ) C ˜ , and iD C µ HW D D 3 † 6 H (6) m S I 3 2 ( = 3, the number of colors. ˜ + ln H τ − H (6) νW ˜ ) H H 2 c H d dC 2 2 )( CL ) ], can for instance be induced by integrating out sterile R O ψ = e N ν ψ ψ R , µ 53 S (5) (6) LdQν , γ µν ¯ γ Lν can be absorbed in a shift in C R C ( (6) Hνe ν ¯ Lσ 1 2 (Λ) usually referred to as a type-I . The T k (¯ ( O ) and L c O (6) − QuνL ( (6) LνH N C = O mn S (2  γ / (6) νW (6) Hνe (6) (6) are defined as the linear combinations LνH S kl SMEFT dim-6 operators [  O  1) O O C ν s Class 3 Class 2 Class 1 π (6) T . α 4 − = C 2  c L (5) ν N = L and = ( µ Table 1 (6) F S (6) QuνL ln C C d dC We evolve the operators from Λ to the electroweak scale using one-loop QCD anomalous We are mainly interested in the operators that appear at dimension-6 [ The first operators have dimension 5 Operators with a left-handed neutrino also contribute to the same observables we discuss here, but the 1 dimensions. The operators The remaining three operators evolve simply as [ and where contributions are suppressed by small heavy-light neutrino mixing angles. interactions is leftpurposes, for the future operator work.Higgs phenomenology The was operators discussedoperators are that, in in presented ref. general, [ insuppressed have if arbitrary table minimal flavor indices flavor violation although is certain assumed couplings [ can be where neutrinos with masses of second operator, for ourthere purposes, appears can a simply dim-5 be transition absorbed dipole in operator, buton we will operators not that consider itoperators involve here. a to single those that right-handed lead neutrino to hadronic processes at tree level. A more general set of At lower energies,respectively, after Majorana electroweak mass symmetrythe terms breaking famous for these Weinberg active operator operators and [ contribute sterile to, neutrinos. The first operator, JHEP03(2021)148 =0 the L (2.9) (2.7) (2.8) (6) ∆ } (2.10) 3 L for the , .. 2 } , c , R . 1 ν  { indicate the + h } (6) VR = (6) LνH ¯ c 3 , . . . , n o , µ 1 C k R 2 γ ] and here we use { 2  , ν 2 . v instead for clarity) R 1 51 c = ¯ e . { (6) T l − only contain a single ¯ c R . , = ν d c + h . µν Y µ ., † =0 σ c  γ e, µ, τ R # . L i, j L e ν R 2 + h ¯ (7) ∆ e v u D √ L o R + h = 0) and is given by + ¯ M d VM R o = L † ν L i µν c ν R where µ  ] R D and ν σ ν , + ¯ L D 51 ←→ (6) νW i u (6) † VL D R =2 (6) VL ijkl C ¯ , ν ¯ c L C + ¯  (7) VL µ R µ (6) ∆ -invariant EFT. The tree-level matching , ¯ c γ R γ v VM L ,M M ν 2 L L R em (6) g =0 LdQν ¯ ¯ e e e c R (6) √ νW L (5) (6) SL R ¯ C ν L L 4 ¯ c + ¯ C (7) ∆ ¯ 2 d d 1 2 2 M L – 5 – U(1) v v µ µ L L − 2 ¯ , e 2 ν γ γ + v g × + † + L L L V √ L c d  ¯ ¯ u u + 4 (6) VL ν R c =2 n n L R (6) Hνe µ L u − v F F ¯ (6) (6) γ 2 LνQd duνe C (6) ∆ M 2 M 1 + ¯ L 2 G G C C √ c L L ¯ e √ v 2 2 2 V =  ¯ R ν h 2 v + ν − 2 1 2 R L ), and the Lagrangians = = − " v −  d (6) SR =0 µ 1 ¯ c 2.7 = = = = L − =2 =0 γ L (6) ∆ L L the generation of the active neutrino, while L ¯ e ¯ (6) ∆ (7) ∆ L u (6) SR SM (6) VL (6) VL (6) VR R ¯ c ¯ c c ¯ . In these expressions we have suppressed flavor indices on the Wilson c ,M L L n L + (6) VL d µ c F L 2 (5) = u D ←− G √ C 2 +¯ for − L 2 µ v } = SMEFT operators up to dimension-7 were given in ref. [ 3 D − ν , contains dimension-four and lower operators involving light SM fields. 2 = =0 = , L 1 µ L (6) ∆ { SM D L L M ←→ = l The explicit matching relations are given by ref. [ At the electroweak scale, we integrate out the heavy SM fields (W-, Z-, and Higgs- for the mass terms in eq. ( and remaining Wilson coefficients involving sterile neutrinos. where coefficients. Each Wilson coefficientgeneration carries of indices the involvedgeneration up-type of and the down-type charged quarks, lepton respectively, (we will often use the labels term each For the operators in table where includes dim-6 operators that conserve (∆ and ) andrelations match for to a SU(3) a subset of these results. We obtain JHEP03(2021)148 ¯ n is × N n (2.14) (2.12) (2.13) (2.11) would bosons, N  W R e . We use a ¯ ) T w ) 3 identity matrix in θ c R R 2 R × , u d ]. To ensure that , ν  !  L . This operator involves (sin w † ν ∗ R D w µ 60 the 3 θ , θ γ (6) νW , 2 M M 1 2 R = ( 59 ) O e [ n † L D sin sin is the Weinberg angle. N + ¯ 3+ 2 3 M M 1 3 νγ , w L †

R − θ e  and →  ) or an insertion of a lepton mass by µ  = γ µ V, n w VM ν (7) VL N γ † θ R V, , . . . , m c ), we also include the effects of the SM ¯ 1 d  2 R (6) , νW u (6) ) is the diagonal matrix of charged lepton νW m 2.8 + C τ C = 3 + + ¯ v denotes the contribution from the SM weak L (6) 2 QuνL + sin 2 n d (6) v LdQν L g , m C .,M 2 diag( 1 2 √ – 6 – u µ g (6) VL C   c 4 . c 2 √ 2 w −  ≡ 8 v 4 , θ v − field which, after integrating out the , m w  ν 2 e θ is strictly constrained because it generates neutrino + h µ ) = = = =  m 2 denotes the CKM matrix, γ j m L sin N ν L ]. Additionally, if these constraints are avoided W = (6) ν νW ¯ V e µ sin (6) SL (6) T (6) 1 3 VL (7) VL ¯ ¯ c c ¯ c γ ¯ C ( 2 3 C M U 58 j L + , c i L ν ν in the following. This effectively implies we do not consider ν ¯ − 1 2 )¯ N 55 = diag( M µ ij 1 2 1 2 − γ T e δ . i (6) νW L   − U ¯ M ν − µ (7) VL O µ c = F γ γ (2 2 L L G symmetric matrix with ¯ m ¯ 1 4 u d 4 √ L ¯ n and ¯ − + ¯ + + × = (6) VL n ¯ are only induced by the dimension-6 operator C is a ¯ VL (7) are the flavor indices of active neutrinos and c (6) neutral ν L i, j M and ¯ Besides the charged currents listed in eq. ( The first term in the expression for (6) VL ¯ where unitary matrix, U, to diagonalize the mass matrix where 2.2 Rotating toAfter the electroweak neutrino symmetry mass breaking the basis neutrino masses can be written as weak neutral currents that contribute to decay processes of sterile neutrinos using the equations ofdipole motion. moments at one-loop [ decay relatively fast into twolong-lived, body we suppress final states via the effects of interaction. All otheradditional entries insertions arise of from leptonicC the mass dimension-6 matrices. operators,a in The derivative some acting operators cases on withleads with a to Wilson operators charged coefficients involving an explicit derivative (¯ for the remaining operators. Here lepton flavor space, and masses. JHEP03(2021)148 l (2.18) (2.19) (2.20) (2.16) (2.17) (2.15) indicate SMEFT ν ν } 3 , . (6) VRR 2 } , , , , C ¯ n ∗ 1 ∗ ∗ µ { γ U ) and neglect the U U s s s ν R = P P P ¯ e ,..., ν , 1 2.18 ν . R ) (6) VL (6) SR (6) T (6) TRR { ) ∗ d c c c C , U µ i, j, k s = ¯ = ¯ = ¯ γ µν  PU P R ( σ n ( u . L × R L ¯ (6) e (6) SRR n (6) VLR TRR P P + ¯ I where R is the usual PMNS matrix. We i is only induced by the ), with that appear in the Lagrangian as = = (6) d TRR 3 ν c × c C c L U R µν ν 2.4 n ijkl σ 0 (7) ν . VLR = (6) = L VLR  ν C u C c m (6) T = µ ) become ), ., ¯ + ¯ νm γ N c s . whose low-energy effects are suppressed by 1 2 ν R + 2.9 e 2.11 m − + h (6) + ¯ SLR )–( (7) VLR N ,C – 7 – C /  ν ∂ν C ,P 2.8 ≡ ν L ¯ νi µ ¯ e  SLR ν 1 2 , U,C (6) n VLL L s D ←→ × d C i = P 3 µ R (6) SRR 0 ν γ ν , ν u (6) VL C ) 3 L L ¯ ν , ν (7) ∗ VLR ¯ e C + ¯ × ) ,C projector matrices . ,C = h 3 C U ν ∗ ∗ + I ∗ s ¯ L n L  U P d (6) U PU ¯ S U e s ( ( s × s µ (6) SRR C L = P L R P PU γ P n d C P P L µ P ¯ (6) VR (6) SL (7) VL (6) VL L u γ . In absence of sterile neutrinos, c c c c ¯ e = =  L m and R L ¯ R u = ¯ = ¯ = ¯ = F 2 d ν ν ¯ n 1 v G L √ UN 2 ¯ u + × (6) (6) SLR VLL (7) VLR (6) which is strongly constrained by other probes. VRR = C C = C C N (6) νW 7) , C mass (6 L . Furthermore, as discussed below eq. ( In what follows we only consider the dimension-six terms in eq. ( Finally, we evolve the operators down to the bottom or charm mass. The vector /v b,c dimension-seven operator proportional to m operator currents do not evolve whilethe the same scalar way as and the tensor scalar dimension-6 and and tensor -7 currents couplings in evolve eq. in ( Each Wilson coefficient again carries fourthe flavor generation indices of thenow involved denotes up the quark, particular down neutrino quark, mass and eigenstate charged and lepton, runs respectively. from where In the mass basis, the operators in eqs. ( to express the relation between the neutrinos in the flavor and mass basis as write the Majorana mass eigenstates as We introduce 3 and define JHEP03(2021)148 the = 1. U 4 e mesons. U s B -, and Higgs Z -, , and 0 W B ,  B , s D ]. From the plots it is clear is the CKM matrix and , 0 63 V D ,  we depict a selection of branching D 1 mesons, respectively. Branching ratios , where 4 s k B U , ij − are similar and not shown to not clutter the V B 2 – 8 – would dominate because of phase space suppres- 0 = 4). The production then arises solely from the −  B k n l = + or , and 4 s 0 N ijk = 1 ( D ] for recent discussions, and here we confirm (most of) , both leptonic (solid lines) and semi-leptonic processes ) D , n N → − 64 (6) m VLL ,  ij D (GeV) neutrinos, mainly because of their smaller production C 63 M O Ψ, and Υ mesons although they would allow to probe a larger ]. and set J/ N , 62 ) with ( , c m B 61 , 2.18 33 We consider leptonic and semi-leptonic decays of 3.2 Sterile neutrinoFor production higher-dimensional from operators higher-dimensional the operators unknown quark in flavor contrast structure tobetween of processes the the involving minimal different Wilson quarks. case coefficients As where is such, each the flavor CKM structure is matrix independent provides the relation plots too much. ForAll these branching examples ratios we are consideredthat, a in depending final-state excellent on agreement the and(dashed, with mass dotted, set ref. and [ dot-dashed lines)state must pseudoscalar be and included vector and mesons. the latter involve both final- For semi-leptonic decays, wedecay consider rate final-state formulae pseudoscalarthe and appendix. and the vector associated In mesons.ratios the decay for left constants decay The and processes and rightfor of form panels analogous of factors decays figure are of given neutral in we require the number ofratio mesons to final produced states at includingwe the a calculate various sterile experiments the neutrino. and latter. thein The For former branching the minimal is literature, models, discussed see these below,these while branching refs. results. here ratios [ have been calculated first term in eq. ( lepton mixing matrix. Athat leptonic broad meson range decays ofsion processes associated are to relevant. semi-leptonicmasses Naively decays, in one but the might amplitude CKM think expressions factors change and this picture. powers of To calculate meson/neutrino the production rate, 3.1 Sterile neutrinoWe production begin in by minimal discussing models trinos sterile interact neutrino with production SM inneutrino fields minimal with only mass models via where mixing. sterile For neu- simplicity, we consider a single sterile neutrino mass range. Production viacontribution the from decay vector of mesons pseudoscalar oflifetime mesons the of dominates same over the the quark latter. composition,bosons Sterile due is to neutrino subdominant the production for muchcross via shorter sections direct [ decays of In this section we discussperiments. the production For of concreteness sterile we neutrinos considerparticles. at the collider case We and where consider fixed-target the ex- theteraction sterile points, production neutrinos containing are through a Majorana the singlecontribution decay charm from or of bottom mesons, quark. produced We at neglect the the subdominant in- 3 Production of sterile neutrinos JHEP03(2021)148 ¯ 4 f B e → N + N + N + N + N + N + N + N + 0 0 0* + +* s - 0 + B e + N + π + ρ + D + D + K + K + D + * s ------f = 21 e → e → e → e → e → e → e → e → D → s - - - - - s s s + B B B B B B B B B } ν ijkl → { N (left figure) or 5 ). All branching = 0. In addition, D = 1. 4 τ 4 and 4 2.18 e e 4 U vector operator has a U = 4 = 13 3 (6) VLR µ } [GeV] C U N , respectively, if the Lorentz m e 2 ijkl + { )), which leads to N 1 6= 0 and 4 → 2.12 e U 0 D -4 -5 -6

10 10 10

mesons). For the plots in this section, we

0.100 0.010 0.001

) + -> ( N B BR s is a pseudoscalar or consisting and D . N + – 9 – N + N + e N + N + X 0 0 0* 0 ′ 2 N + π + K + K + N + N + η + η + ϕ + N + K + + ------. For concreteness we consider decays of Majorana and e → e → e → e → e → e → e → e → e → 4 - - - - s s s s s N s k D D D D D D D D D U B where → ij , depending on the sterile neutrino mass, either leptonic X V 1 B 2 2.0 + − e = + 4 1.5 . We consider the flavor structures N ijk ) → (6) VLR (6) VLL C [GeV] 1.0 D N C denotes any SM that is kinematically allowed (in case of quarks, m f , and and ) is ( 0.5 X . Branching ratios of sterile neutrino production channels through (6) TRR 2.18 + C , e The three chosen operators correspond to quark bilinears with different Lorentz struc- Here we discuss a few cases only. We consider the operators with Wilson coefficients 0.0 1 -4 + 10

0.001 0.100 0.010

(6) SRR ) + -> ( N X D BR in eq. ( sterile neutrinos into final-state electronswe and consider set the SMdecays weak where neutral current (seewe eq. consider ( a final-state neutral meson). These decay rates have all been calculated in the 4 Decay of sterile neutrinos 4.1 Sterile neutrino decaysWe begin in by considering minimal the minimal models scenario, where we assume that the only non-zero term modes are those withsimilar a Lorentz final-state structure vector as meson. theture. Finally, SM the As charged was weak current, the(solid but case lines) with or in semi-leptonic different figure processes flavor (dashed,the struc- dotted, production and of dot-dashed lines) sterile can neutrinos, dominate and must all be included. tures (scalar, tensor,allowed and and vector, these respectively). dominatethe over sterile In semi-leptonic neutrino the decay mass.forbidden scalar modes and For case, for a the all leptonic final-state tensor meson considered decays operator, must values are however, be of the produced. leptonic In decay these mode cases, is the dominant decay structure permits this. TheseN choices also allow forof semi-leptonic just decays up, of down, the anda form strange strange quarks quark (strange as quarksassume is the only weak the if interaction the case is decayinga for turned meson time. off contains and The consider results only are one depicted non-zero in EFT figure operator at sponding to several flavor structuresratios for each can effective be operator calculated in from eq. the ( expressions given inC the appendices. that allow for leptonic decays Figure 1 (right figure) mesons in the minimal scenario for final-state and unless model assumptions are used. This leads to a large number of possible cases corre- JHEP03(2021)148 N + N + N + N + N + N + N + N + N + N + N + N + 025 0 0 0 + +* + +* + +* 0 0 0 . (4.1) N + π + ρ + N + π + ρ + π + ρ + K + K + K + K + K + K + ------e → e → e → e → e → e → e → e → e → e → e → e → e → e → - - - s s - - - s s - - s s = 0 B B B B B B B B B B B B B B Ψ). This / 4 e ]. 21 ,J ) ∗ s 64 D 5 5 5 (6) , TRR ∗ C D 4 4 4 , 1, ( φ . , , ∗ 3 3 3 )) = 0 =0.1 =0.1 K =0.025 [GeV] [GeV] [GeV] N N N VLR SRR , D m m m 4 TRR C C e C ω 2 2 2 + 21 , ) ρ u + ¯ (6) 1 1 1 SRR 1, respectively. τ . + ) C ν e ν = 0 0 0 0 → 1 1 1 -4 -5 -6 -4 -5 -6 -4 -5 -6 4 e

10 10 10 10 10 10 10 10 10 τ

→

0.100 0.010 0.001 0.100 0.010 0.001 0.100 0.010 0.001

) + -> ( ) + -> ( ) + -> ( N X B BR N X B N X B BR BR ( . For heavier sterile neutrinos other multi- 13 τ ) ρ decays tree Γ( N + N + N + mesons in the left and right panels, respectively. N + N + N + N + N + N + N + N + N + – 10 – Γ (6) τ 0 0 0 0 0* 0 0* 0 0* VLR 0 0 0 > m N + π + ρ + N + π + ρ + π + ρ + K + K + K + K + K + K + C ------B ≡ e → e → e → e → e → e → e → e → e → e → e → e → e → e → N ) - - - s s - - - s s - - s s D D D D D D D D D D D D D D τ ) and vector meson ( m c and m η ( , D s 025 and ( . ]. Most results agree with each other and with our findings D QCD , 68 = 0 – D 4 e , ] and estimate the total hadronic decay width by calculating the 0 leptons through both charged and neutral weak currents. The 63 1.5 1.5 1.5 1 + ∆ 13 ], assuming η 63 ) , 63 → 1, respectively. Right: from top to bottom the figures correspond to η . , (6) TRR =0.1 =0.1 N 1.0 1.0 1.0 =0.025 C [GeV] [GeV] [GeV] K N N N VLR SRR m m m = 0 TRR , C C C 1, ( π . 4 e mesons [ 21 0.5 0.5 0.5 = 0 ) ρ . Branching ratios of 4 e (6) VLR 13 ) C We consider 0.0 0.0 0.0 -7 -4 -5 -6 -7 -4 -5 -6 -7 -4 -5 -6

10 10 10 10 10 10 10 10 10 10 10 10 (6)

SRR

0.010 0.001 0.001 0.010 0.001

) + -> ( ) + -> ( ) + -> ( N X D BR N X D BR N X D BR C decay width to spectatorare quarks taken times from appropriate a loop comparison corrections. to The hadronic loop corrections pseudoscalar ( effectively also takes intodecays account of decay modesmeson into final two states become throughInstead relevant the and we follow intermediate summing ref. exclusive [ channels becomes impractical. literature, see e.g. refs.given [ in the appendix, with thewhere exception for some decay differences processes appear. into final-state For neutral these mesons cases, our results agreelatter with leads ref. to [ the invisible three-neutrino decay mode. We include decays into a single Left: from topand to ( bottom( the figures correspond to ( Figure 2 JHEP03(2021)148 5 (4.3) (4.4) (4.2) , as a N ¯ qq ], apart cτ → 4 63 2 N e 1 U is the proper )] Γ 0.50 N N τ [GeV] m N ( m while the branching 0.10 QCD , 4 0.05 ) 1 GeV, while the decay to qq . + ¯ , which shows branching ratios e N 0.01 3 6 1 -6 /ν m 10 + hadrons)

10

1000

..., 0.001

−

e4

] [ τ m c N e

U 1 GeV) [1 + ∆ 2 e + /ν − 2 s 2 → − π α is the speed of light and e N 1 GeV are shown in the right panel of the s c N 2 c D + . m ( > * s ( → θ ). Right: Decay length of the sterile neutrino in N tree + 5 almesons all → N N D → N →π → N ,η',η → N ρ → N ω+ϕ → N + Γ m s – 11 – 4.3 Γ( π α ≡ = ) 5 = 5 GeV, the single meson final states make up roughly N m N QCD 4 single meson ( ∆ m → N QCD , we show a plot of the scaled proper decay length, 3 )Γ quark and 3 N . s m 1 + ∆ 1 GeV are shown in the left panel of figure or [GeV] 1 GeV. At − N d < m 2 leptons & → N in the minimal scenario, where N N m m N . The branching ratios to individual mesons, leptons, and three neutrinos (1 GeV +Γ θ m N decay rate and we assume this to hold for sterile neutrino decays in the minimal = denotes a τ . Left: Comparison of branching ratios to individual mesons divided by the inclusive N D Γ 1 1 We write the total decay rate as We find that single meson channels dominate for

0.05 0.01 0.50 0.10

igemeson single quarks Γ / Γ function of lifetime of (invisible) for ratios to quarks, leptons, andsame invisible figure. for In the right panel of figure significant. We demonstrate thisto in individual the mesons, left compared plotto to of quarks, the figure for sum of20% all single-meson of final the states, hadronicfrom and decay decays compared rate. to neutral Our vector results mesons, which are only in play good a small agreement role. with ref. [ to calculate the inclusiveneutral hadronic weak sterile currents. neutrino decay rate through bothquarks charged become and relevant for larger masses, indicating that multi-meson final states become where dots denote higher-orderhadronic corrections. This gives ascenario good as description well. of That the is, inclusive we use where Figure 3 hadronic branching ratio calculatedminimal via scenarios. eq. ( JHEP03(2021)148 τ and B quarks → N leptons → N invisible → N = 0. Left: 4 5 τ U 4 = 4 µ U 3 [GeV] N m 2 1 GeV and the decays to quarks = 1 and > 4 e N U m we plot the proper decay length of 5 1

, but we treat the mixing angle as a free 0.4 0.2 1.0 0.8 0.6 BR N = 11 we consider decays into a single charged /m – 12 – K + ij ν * only pions are relevant. As is the case for the m 4 invisible → N leptons → N π → N ,η' → N K → N ρ → N e ∼ with 11 ) 2 4 | 4 1 ije e (6) SRR ) involve two quarks and a charged lepton and induce new ) U | C (6) VLR 2.18 C 0.50 ). We always consider the case of a single sterile neutrino which is 2.18 [GeV] N m 2 GeV for scalar operators, as we do not have the benchmark of hadronic & meson, while for ( 0.10 N . Branching ratios in the minimal scenario for ρ m , against its mass for various choices of Wilson coefficients and flavor assignments. 1 GeV with decays to individual mesons. Right: sterile neutrino (a 3 + 1neutrino model). only Because interacts of through minimal3 charged + sterile-active 1 mixing, and the minimal neutral sterile seesawsterile SM model, weak neutrino the interactions. masses mixing In angle the is related to the ratio of active and N Here we consider the minimal scenario without higher-dimensional operators and 1 < 1 0.05 cτ mesons. We consider three classes of scenarios that are representative of the effective 1.

0.001 0.500 0.100 0.050 0.010 0.005

, N BR In this work we focusD on the productionLagrangian of in sterile eq. neutrinos ( throughmixed the with decays the of active . The three classes of scenarios are listed below: N We consider one effectivecoefficients coupling are at set the to time, unity. turn off minimal mixing,5 and the Wilson Theoretical scenarios decays to verify our resultsmesons. for This non-SM leads currents.consequently to We renders only a our consider potential sensitivity decays underestimate limitsdecay into for of individual length future the experiments regime. sterile conservative insignificant neutrino As the decay deviations all large width from our and our results findings. below are In given figure on log scales, we do not expect or minimal scenario, one can imagineneutrino multi-meson masses states to (for become thisdo relevant flavor for not larger choice, include sterile such such statesaround states here, would although be we find three- that or decays more to quarks pions). become dominant We 4.2 Sterile neutrino decaysThe from EFT operators higher-dimensional in operators eq.channels ( for sterile neutrinosof to the decay operators, into typicallyinstance hadrons. for only an one Depending operator or on ( two the single-meson flavor decay structure modes are relevant. For Figure 4 m correspond to the total hadronic branching ratio. JHEP03(2021)148 (5.5) (5.1) (5.2) (5.3) (5.4) , to be , LQ 1e4 11 e4 12 e4 21 1e4 11 e4 12 e4 21 1e4 21 1e4 11 e4 12 ) ) ) ) ) ) ) ) ) ∗ 44 m U (6) (6) (6) (6) (6) (6) (6) (6) (6) VLR SRR TRR VLR SRR TRR VLR SRR TRR 1 (C (C (C (C (C (C (C (C (C ije  ., (6) SR c . ¯ c ., c  . + h = + h 4 Rl 5 . ν  ije a ∗  Rl ˜ R . ν RL jk 6), which can couple to (sterile) b i a Li y / ¯ (6) Q TRR ¯ 1 Q  LR C il (6) LdQν , y  LR 2 il C ab y  2 2 , 2 2 LQ  1 v 3 + j = 4 ( m d 4 are flavor indices, respectively. Note that = ˜ a k R b Lk = ¯ L ije L (6) T  c ab 1  ijkl – 13 – a ijkl  (6) SRR ˜ = 4¯ i, j, k, l  R C R [GeV]  Rj (6) S N ¯ (6) LdQν d ¯ c (6) m LdQν C C RL jk , 0.5  y  4 e − = U ) ij = R V (6) ν 2 L LQ 2.11 − L 0.2 = 4 ije ) are SU(2) indices and  0.1 2.18 ]. We focus on the representation (6) a, b VLL 1 -6 C 18  10 . Proper decay length of sterile neutrinos for various choices of EFT operators and flavor

0.001

= 1 in the 3 + 1 model. LHC constraints force the leptoquark mass, ] [ τ m c N Finally, going to the neutrinoand sterile mass neutrinos, basis we and obtainin focusing the eq. matching on contributions ( the to couplings the to effective operators electrons where We read from eq. ( l above a few TeV andthis for leads to low-energy the purposes effective we operator can integrate it out. At tree level ref. [ neutrinos through the Lagrangian where parameter. We stressneutrinos that and the is minimalsimplicity thus 3 it + ruled provides 1 a out model useful by benchmark. leads neutrino to oscillation two experiments, massless but active the exchange with of its . All possible representations of LQs are summarized in In this scenario we extend the minimal 3 + 1 model by interactions generated by 2. Figure 5 assignments. The Wilson coefficients areoff. set to unity and the minimal active-sterile mixing is turned JHEP03(2021)148 ], el- 29 [ (5.6) (5.7) ij of the V . Instead  W CODEX-b ], and for simplic- , are all designed 36 [ . (6) VLR MoEDAL-MAPP2 0 C SHiP > are generally induced as and 4 and ANUBIS ije ],  , (6) VRL ) in combination with the SM (6) VLL 35 C [ C (6) VLR = 1 2.18 . In minimal left-right symmetric C and 44  N AL3X ]. Of these we shall focus on the search and from leptoquark interactions pro- /m , which is the largest experiment in de- MoEDAL-MAPP1 ν (6) 4 VRR in eq. ( e 72 , , C m 4 U ,U e 71 p ν [ U N (6) ATLAS VLR and the combination of the LQ couplings and ], and ij m m = C V – 14 – N 34 4 2 r – e LHCb m − U ' 32 [ = 4 , this reduces the effective number of free parameters e 4 j U ], and ije at a time. It is straightforward to generalize this choice. symmetry, the dependence of the effective operators on the  28 [ and ]. Currently at the LHC we have the operational experiments ij C can be calculated. We do not consider this here and consider ]. These latter experiments, including i (6) VLL 41 MATHUSLA or – CMS C ij . 38 ],  , ], P 39 2 LQ 37 31 27 . We will use the canonical see-saw relations , [ 05 eV as representative for the active neutrino masses. For a specific . denote the up- and down-quark generation, respectively, and ∗ /m 44 30 ∗ j [ U = 0 RL je ν y ATLAS and m ], 1 LR i i y FASER2 70 , collaboration [ 69 [ and one particular choice of ity we do not consider theseanalysis effective if operators so here. required. Theythe can leptoquark For be the easily scenario added active-sterile and tomodels neutrino set the with mixing exact parameter, wequark flavor follow indices In left-right symmetric scenarioswell. nonzero The resulting phenomenology is very similar to models, with right-handed chargedof gauge bosons, implementing which the can fulland mix consider left-right with the symmetric effects model, ofleft-handed a we weak nonzero interactions. take That a is, simplified we scenario consider and set quark flavor choice of to two: the sterilemass neutrino mass ements of the CKMmixing matrix. proportional to In the thisportional mixing scenario, to angle we have contributions from minimal where at the CERN SPS [ The final scenario we consider is inspired by models, such as left-right symmetric 3. teraction points (IPs), namely inFASER alphabetical order: MoEDAL to specifically lookpotential for of neutral all long-lived the above-mentioned particles experiments (LLPs). specifically for We sterile neutrinos. shall discuss the search ing both existing and proposed experiments,SHiP as well as the proposed fixed-targetALICE experiment sensitivity for long-lived sterile neutrinostector at volume, and can thusseries in of new principle experiments explore has the recently largest been decay proposed lengths. at various Beyond locations this, near a the LHC in- 6 Collider and fixed-targetWe analysis proceed to explore the search possibilities for the above scenarios at the LHC, consider- JHEP03(2021)148 ]. 80 , ATLAS 79 , 38 , detector was 36 , 35 -mesons. This can , detector which has ATLAS mesons, we will use D 33 ,  denotes the remaining . The long-lived sterile B 1 . It is worth mentioning 32 X , ATLAS - and M 4 B 30 and 500 meters away from the IP. , −  29 D in the very large pseudorapidity regime. – 15 – Pythia 8 ] is insufficiently validated in the forward direction, which 74 , . that is to decay into a sterile neutrino, is produced in the process experiments. In fact, even for 73 [ 1 M . Similarly, for the displaced decay of sterile neutrinos, we consider ATLAS 3 , employ a far detector with a distance 5 FASER and decays promptly into e.g. a sterile neutrino ( the rock and shielding below ground should remove the SM background. As Pythia 8 coverage immediately around the IP, and a large irreducible SM background ATLAS ] to correct the behavior of X and as these are only relevant for very light sterile neutrinos, and their π 78 + –  1 π 75 [ M In addition, since the detector efficiency of the future experiments is unknown, in One potential issue relates to possible background events which, depending on the Instead of performing a detailed study considering different components of the de- We do not study the production of the sterile neutrinos from the decay of lighter mesons In this section, we review briefly the setup of the beam and detector geometries for MATHUSLA → experiments except is expected. Dependingvary. on Instead the of signal performingimplement type, an an the estimate estimate number of for of the background efficiency, events such which for background we each events discuss scenario,order below. may we to shall make a fair comparison, we assume a 100% reconstruction efficiency for all the stallation of veto and shielding segments,For as argued in refs. [ for cosmic rays, directional cutsdifferent will experiments, be we applied. show 3-event To(C.L.) isocurves assess which with and correspond compare zero to the 95% background.an sensitivities confidence almost of level This 4 is not appropriate for the placement of thehadronic detector, interaction with may the detector consist material,study, except etc. of All long-lived the experimentsThe SM considered space in between hadron the this IP decays, and the far cosmic detector is rays, usually sufficiently large to allow for the in- tectors separately, for simplicityand we make will a take comparisonnot the between designed whole the to detector various look experiments. asbased for on the Since existing neutral fiducial neutral the LLPs, LLP volume, considered searches, we which, here. shall however, search add for an heavier candidates estimate than for the efficiency factor FONLL Finally, we ignore the vectortypically many mesons orders decays of into magnitude sterile largerbranching than neutrinos, ratios that as for of sterile their pseudoscalar neutrino mesons decay leading production. width to is tiny if at least two tracks are observed stemming fromsuch the as same DV insidesimulation the in detector. is relevant for the multiple hadronic states plusthat a the lepton, heavy meson as explainedpp in section decay products). Such signalwith events e.g. can the be promptneutrinos observed decay lepton at in a accompanying macroscopic the the distance, near where production the detector displaced of such vertex (DV) as is reconstructed each experiment and introduce thesimulation. Monte We Carlo focus (MC) on simulation theproceed procedure production via for either via purely the the leptonic event rarediscussed two-body decay in decays, of section or semi-leptonic three-bodyboth decays, the as two-body decay into a charged lepton and a charged meson, and decays into JHEP03(2021)148 2 1 1 (6.1) − stands − -mesons B SHiP 300 fb 3000 fb , as its beam - and 1 1 sensitivity range. − D − SHiP SHiP 300 fb 3000 fb 1 − IP and the LHC accelerator. , 1 ]. For various other theoretical − 8 m 85 . – LHCb on target (POT) are planned. 0.17% 0.68% 3.8% 30 fb 81 123 ]. It has not been approved yet. With a 20 , or , the sterile neutrino. The active decay < 41 10 100 or 250 fb – N N × CMS 39 cτ , 6 1 1 N − − − γ – 16 – will have a significant forward boost. This will 10 z N 1 × ATLAS M 1 < β . 3000 fb of sterile neutrinos is produced in a hadronic collision 1 1 8 m . M 8 68 1 − POT 3000 fb − project, a total of 2 10 20 experiment is proposed to have a cylindrical detector downstream × SHiP ATLAS AL3X ANUBIS CODEX-b 0.89% 100% 13.73% 1.79% 1% 10 FASER FASER2 MAPP1 MAPP2 MATHUSLA 1 SHiP . 150 fb ] for a recent review. 1 × SHiP 2 86 , the initial meson facility was proposed to make use of the high-intensity CERN SPS beam of . . Summary of integrated luminosities for the various experiments. “POT” for SHiP 2 SHiP At We start with a brief introduction of the fixed-target experiment The search potential of these experiments, but also of other fixed-target experiments, Int. Lumi. Int. Lumi. Experiment Experiment In this work, we consider the LHC center-of-mass energy at 14 TeV for all experiments. We assume an Angular Cov. Angular Cov. 2 sensitivity is for particles with LHC upgrade before theour experiments results. are online. Changing it to 13 TeV would only have a small effect on is strongly forward peaked,be and passed the onto thechamber decay is products, 68.8 m including of downstream 60 of m, the whereThe target. the front first It surface 5 has has m an a are elliptical cyclindrical to shape shape be with with used semi-axes a for of length placing 5 m background and suppression 2.5 m. vetoes. The optimal decays, fly an extendedespecially length, if and the then lab-frame decayFor decay the inside length lifetime the of of detector the the chamber LLP downstream, lies within the between the beam protons and the target material. The differential production cross section (solid) molybdenum alloy and pure tungstencenter-of-mass energy [ of approximately 27 GeV,are large production expected. rates The of at roughly 70 mlong-lived away from neutral the particles, IP. which The are experiment produced is from specifically e.g. designed charm for or detecting bottom meson rare in table 6.1 SHiP The 400 GeV protons incident on a fixed target made of e.g. a hybrid material composed of has been investigated forscenarios, example see ref. for [ [ setup is differentwhich from are the all other associatedSince experiments, to these and either experiments the then differ in discuss the the projected other integrated luminosity, experiments, we summarize them Table 2 for “Protons on Target”. We also list the simple geometric coverage for each experiment. JHEP03(2021)148 ] ]. 78 – 87 (6.4) (6.5) (6.2) (6.3) , 75 [ , . 84 , collisions Pythia 8 15 14 83 experiment, -mesons, we pp 10 10 B FONLL × × , . , it is necessary 16 12 62 53 ) production rate . . ATLAS b 10 10 ¯ is included, and for b - and LHCb ( × × /  = 6 = 2 in its 5 year lifetime: D ¯ c 0 2 D . . c CMS / = 6 = 7 SHiP into s s HL-LHC HL-LHC D B s s sites, we consider ∗ solid angle the following list of ]. The SHiP SHiP D B ATLAS D π 83 ,N ,N . As a result, the mesons and the 16 15 LHCb ,N ,N / 17 13 10 10 -meson production cross sections for a SHiP 10 10 × × CMS B / × × 89 46 . . 3 7 . . = 3 = 1 ATLAS – 17 – = 4 = 2 , respectively. These numbers can be estimated s 0 0 B SHiP SHiP D B -meson and 0 0 N HL-LHC HL-LHC D B D ,N ,N 17 13 ,N ,N , and 0 , the decay branching ratio of 16 15 10 10 B ) per collision. After the fragmentation factors are taken into 10 10 × × N 7 4 7 − × × . . ,   HL-LHC 10 D 04 46 B integrated luminosity over the full 4 ]. See also the earlier work in ref. [ . . = 1 = 2 N × N 1 63 mesons are in principle also produced and may extend the upper reach − 6 ,   = 2 = 1 -mesons the corresponding fragmentation factors determined by . c s SHiP SHiP D B we can extrapolate these cross sections to the whole kinematic range. We are the relativistic speed along the beam axis and the Lorentz boost factor B D IP, the differential cross section for the production of GeV-scale mesons is B (1 N N N ]: 3 N and extended programs at the − , collaborations reported   63 -collider, for certain kinematic range. Using the simulation tools 0 , γ HL-LHC HL-LHC B D 10 D z N pp ATLAS integrated luminosity. To achieve this, we follow the procedure in refs. [ N N β N . However, given the much smaller production cross section, we do not take it into = 14 TeV with equal beam energies. These experiments benefit from a beam energy × 1 LHCb , N , respectively. s Pythia 8 7 To perform the sensitivity estimate for experiments at In order to study sterile neutrinos produced from the decays of ATLAS − .  √ m N D by the integrated luminosity. 6.2.1 FASER At the strongly peaked at large pseudorapidities, i.e. close to the beam pipe in both directions. For the estimate of the number of are used. These numbers willbut be also used for not only all for the the extended evaluation at programs the discussed below with a possible overall re-scaling and find that for 3 ab numbers of the produced charm and bottom mesons: to know the inclusive production rate3 of ab the heavy-flavor mesons atThe the HL-LHC with up to 13 TeV For at that is orders oftherefrom produced magnitude sterile higher neutrinos compared areeven for much to more detectors boosted, that are leading to to be good installed sensitivities hundreds of meters away from the IP. We note that in account. Similarly for the other LHC experiments,6.2 as discussed below. Experiments at the LHC is 1 account, the numbers of thefrom ref. heavy-flavor mesons [ can be estimated and reproduced below of need to know theN total number ofby these following mesons ref. expected [ at where JHEP03(2021)148 1 − ] in CMS ∞ FASER FASER2 + , integrated 17 . 1 and − [9 ∈ . η FASER and is tilted roughly at . The pseudorapidity 2 5 m, and the expected . . m is not well validated in 4 FASER 100 m and a height of 25 m. 1%. In a more precise Monte Carlo × . 2 Nevertheless, it has been shown ≈ ) Pythia 8 5 2 ] while the azimuthal angle remains , when only the is 9 m ], ∞ − 5 84 + 10 10 , · 2 86 . ∼ . (2 2 [6 https://faser.web.cern.ch/ | / : ” (MAssive Timing Hodoscope for Ultra Stable ) 4 ∈ ◦ e – 18 – U η | FASER ] has been proposed, which is designed to specifically https://mathusla-experiment.web.cern.ch/ cos(63 . 2 is currently planned to be upgraded to a larger version It is placed in the existing TI12 tunnel at a distance 31 , MATHUSLA m 3 4 . The corresponding angular coverage is detector has an base area of about 10 30 1 FONLL − FASER detector is at about 132 m from the IP. The area of the enclosing sphere in the azimuthal angle. IP and has a cylindrical shape exactly aligned with the beam π MATHUSLA from the IP still leads to a small geometric coverage. It corresponds It is proposed to be built for the HL-LHC phase with 3 ab IP, the front edge of the detector should be horizontally shifted along ATLAS 4 MATHUSLA ]. . The , to be under operation during the HL-LHC era, collecting up to 3 ab 2 CMS 34 m – ]. 5 32 10 34 MATHUSLA · ], and located in the same place. Its geometry is specified as a cylinder with 1 m at different ranges of the transverse and pseudorapidity by using the FASER2 2 . 31 detector has a cylindrical radius of 10 cm and a length of 1 As mentioned earlier and discussed in ref. [ After Run 3 is finished, relative to the radial direction. 10 See the official websiteSee of also the the experiment webpageThe of center the of experiment: the 3 4 5 ◦ is about 2 63 integration we found a coverage of 3.8%. to a solid angleto or be geometric one coverage ofsmall of the active-sterile about far neutrino detectors 3.8%. mixing atconsidered at the [ LHC that may have the strongest reach in the very neutraL pArticles) hasexperiment been [ suggested toluminosity. be With constructed a at boxRelative shape, the to it surface the has above athe the base beam area axis of bydistance 100 m 68 of m, and vertically upwards by 60 m. Despite its huge size, the large 6.2.2 MATHUSLA A much larger experiment called “ the large pseudorapidity regimebottom for the mesons. differential production ToPythia cross solve section 8 this of charm issue,more and we reliable results re-scale given the by meson production cross section in known as of data [ radius and 5 mcoverage length, is which correspondingly is enlarged 300x to fully larger covered. by In volume, thisfor than work, sterile we neutrinos will as study neutral the LLPs. search potential of both collision axis, but slightlyFASER off of theintegrated beam luminosity is due 150 fb topseudorapidity the and curvature full of 2 the accelerator. The then also be peaked(Forward Search in ExpeRiment) the [ largemake pseudorapidity use regime. of this A feature.data far It during has detector been Run known officially as 3of approved by of 480 CERN m and the from should LHC. be the collecting The production rate of light neutral LLPs, resulting from the decay of the mesons, should JHEP03(2021)148 is or 7]. 0). . ) = x and , 3 b 0 , 1 , 9 , z . − b [0 ATLAS , y 25 m from b ∈ x experiment. ) = (0 ∼ η ], respectively. . With a larger 87 3 LHCb x, y, z (AN Underground integrated luminosity. ] to be built at the LHC 1 − 35 ANUBIS is along the beam axis, 6] with the azimuthal angle . , currently under deployment z 0 3 , 2 ] is designed as a cubic box of . [0 29 130 m ∈ from the beam. During the HL-LHC ∼ is predicted to have higher sensitivity. η ◦ , and designed for searching for neutral 14 m), where , LHCb MAPP2 – 19 – 80 m , integrated luminosity from the LHC is projected. 1 − experiment will have a significant upgrade in a second ) = (0 t ] to construct a detector, named , z t is vertically transverse, and the IP is at ( ] at the IP8 of 36 MAPP , y y of data. It is designed to occupy almost the whole of the UGC8 t ] for a sketch. MC integration finds the angular coverage of 38 , 1 x 10 m. Occupying an empty space with a distance − 37 62 , the × , ] proposed two values of the integrated luminosity, 100 fb ]. experiment sits. Placed at a horizontal distance along the beam line 36 35 87 10 m FASER × 4%. The total geometric coverage of the solid-angle is about 1%. 79% [ program is planned to be under operation at IP8 until the end of Run 5, is a rather small detector of volume . . ALICE 6 experiment [ ∼ IP, it covers the pseudorapidity range of (COmpact Detector for EXotics at LHCb) [ integrated luminosity. MAPP1 14 m) to the top ( MAPP2 , in order to accommodate practical concerns including the move of the IP, beam to be 1 , 1 1 − − (MoEDAL’s Apparatus for Penetrating Particles) is proposed as one sub-detector of (A Laboratory for Long-Lived eXotics) was proposed in ref. [ IP. Consequently, a total of 3 ab LHCb MoEDAL 24 m , is 0.85 m (5 m) withThe the length authors 12 m. of This ref.250 gives fb a [ pseudo-rapidity coverage of quality, and investigation of background250 events. fb Here we focus on the benchmark value of 6.2.6 AL3X AL3X IP2 where the of 5.25 m from the IP,beam the detector line, has corresponding a cylindrical to shape a aligned full with and azimuthal surrounding coverage. the The proposed inner (outer) radius era, the accumulating up to 300 fb gallery in the LHCintegrated tunnel luminosity complex, and taking aThe up bigger two a volume, detectors volume cover of about about 0.17% and 430 m 0.68% of the total solid angle [ the LLPs. Similar to version. and expected to collectIt data during is LHC roughly Run 55 m 3 from with the up IP8 to at 30 fb a polar angle 5 the coverage of 6.2.5 MoEDAL-MAPP MAPP 6.2.4 CODEX-b An external detector extension hasCODEX-b also been proposed atdimensions the 10 IP8 m of the 56 m. The axis(0 of the cylinderhorizontally runs transverse, from and thePlease middle refer point of to the refs.ANUBIS bottom: [ ( It has recently been proposed [ Belayed In-Shaft search experiment), inCMS one of theSimilarly service to shafts the just detectors above discussedaxis the above, oriented it also in has the a vertical cylindrical shape direction. however with The the cyclinder diameter is 18 m and the length is 6.2.3 ANUBIS JHEP03(2021)148 = 2 is the 2) as the / β 1 is to be − ], we found is in a shaft + (43 83 2 )] of the LLP’s 11 p MATHUSLA βγcτ ANUBIS ( in angular coverage, , and assuming zero / 1 for the LLPs produced π − is a well-established ex- -mesons. 24 m D − SHiP ATLAS exp[ − offers almost 4 , designed as a fixed-target experiment is far down along the tunnel, but still close SHiP ] for a related study on dark at LHCb. , of course). ATLAS 88 – 20 – CMS FASER (or ] some of us considered the search for light long-lived with the other experiments we must impose a realistic 83 for the scenarios we are considering here. We discuss ATLAS CMS ATLAS , as well. See ref. [ can study decays upto a length of about experiment, which is by volume the larger experiment. In principle this or CMS detector. It has a cylindrical shape with inner (outer) radius of ATLAS denote the LLP lifetime and boost factor, respectively, and ATLAS ATLAS γ is competitive with or slightly better than 6 . ATLAS . We thus furthermore assume 100% signal efficiency. and τ ATLAS ATLAS all events are of interest. With respect to light long-lived particles there are FASER -mesons decays, and was somewhat worse in the case of B All this does not hold for As we have mentioned, all the above proposed new experiments are specifically designed Here we describe in more detail the geometrical parameters we use for the fiducial We focus here on the a priori 6 order to compare a searchsignal at efficiency, to takededicated this into study account. at Toseveral our related knowledge analyses at and the estimate a moment, signal there efficiency is basedstudy no on could be this. performed for periment, operating in theas immediate vicinity of thethus IP. large There backgrounds is which must purposelythe be background, no will dealt shielding, also with. affect the Any signal cuts efficiency, which most are likely imposed in a to significant reduce manner. In which has minimal overhead shielding. to the beam pipe.they In can order tackle to theThe compare issues precise these concerning design experiments background, we ofphase and have these of assumed for detectors zero now is assumed background also that for not all. yet well-established, except for the first to look for lightshielded long-lived from particles. many SM Theyseparate backgrounds are generated background thus at further issues, the awayinstalled IP. depending from All on the on the the LHC same where ground IPs they and they will and thus are all highly have located. susceptible to cosmic rays. spectrometer), and a length oflong 43 as m. its In distance principle,of from even displaced the if tracks, IP a it is DV canto is larger be inside include than the reconstructed. the only beam detector However, DVs pipe,benchmark to spatial that value as be resolution are for more and the located conservative, consists integrated we inside luminosity choose the over the detector lifetime volume. fo We the take HL-LHC. 3 ab that for 100% signalbackground, efficiency, an integratedfrom luminosity of 250 fb volume of the 0.0505 m (11 m) corresponding to the beginning of the inner detector (the end of the 24 m, where 11 mlength, is as the we cylindrical discuss below radiusdecay in and more where 24 detail, m the is fractionrelativistic half 1 the speed detector of length. thewhich Over particle. is this significantly However, larger than the other detectors at the LHC. In ref. [ In the previous sections we havedesigned discussed to proposed look for new long-lived experiments particles. whichthe are They various specifically are IPs typically at placed some thewith distance a LHC away long from or, decay in path. theparticles In case e.g. at of ref. [ 6.2.7 ATLAS JHEP03(2021)148 , 4 − (6.8) (6.6) (6.7) 10 · considered ]. A Higgs searches. In detector. For point of view 90 , geometries of N ATLAS can then be esti- , . ] ATLAS ATLAS cτ ATLAS ], in order to extract dec 92 N N , produced in the initial 74 i signal) , M 73 → [ , ) N to all the X in a given scenario. “Br” stands finds a signal efficiency of at best 3 + , Br( − N N i · → ATLAS in f.v.] i i M in f.v.] Pythia 8.243 N [ N P Br( [ · P – 21 – i MC N =1 i X · h M N N MC 1 i N prod N X searches for pairs of produced particles. This reduces N N . = is the number of initial mesons, = i i ≡ M SHiP prod dec N N ATLAS N N N considered the following scenario proposed in ref. [ considered the following scenario. A decays to a pair in f.v.] is found, similar to the other analysis. In ref. [ N , produced as [ 5 P N ATLAS − h ATLAS 35 mm, and dropping off rapidly for smaller or larger 10 ], ], · ≈ 89 91 cτ is summed over all mesons that can decay to i for M 5 The number of sterile neutrino decays in a detector volume In all these analyses, In ref. [ In ref. [ − 10 · We use the expressions mated by taking into accountthe the detector, boost and factor the andparticles. decay traveling branching direction For ratio of the of latter,state, sterile we which neutrinos contains consider into solely all the three the signal neutrinos, decay and final-state channels is except mediated for by the the SM fully weak invisible interaction. where for decay branching ratio. collisions at the LHC or for 6.3 Monte-Carlo simulation We perform the MCthe simulation kinematics with and the tool tosterile estimate neutrinos, the number of signal events. We express the number of background but also efficiency.somewhat Overall optimistic we flat haveaddition signal applied we a efficiency assume from that factor the the of corresponding 10 cuts reduce the background to zero. the lightest mass scenario theefficiency Higgs of boson 3 is 125a GeV related and scenario the scalar witha is scalar 8 masses scalar GeV. down A of to detector somewhat 5 5 GeV better GeV. with than For a in a Higgs the decay other boson studies. length of 125 of GeV, 75 and cm they find a signal efficiency of 7 would have a muchand higher the boost purely hadronic than decay5 our of sterile the dark neutrino’s. For the lighter Higgsof mass neutral long-lived scalars,and which one in in turn the each muon decay spectrometer, to thus utilizing 2 different jets, parts one of in the the inner detector boson decays to twoan dark invisible , hidden which in lightestand stable turn decay particle. decay to either The one to darkphoton or a masses photons two between pair are dark 0.4 of the photons andmass long-lived , plus range. 3.5 particles GeV or However are the a Higgs considered, boson pair i.e. masses of similar are 125 to electrons/pions or our 800 (light GeV sterile and hadrons). neutrino thus the dark Dark photons JHEP03(2021)148 is 4 (6.9) ]. As MC N and 85 N 3 – Pythia 8 and of the 83 , the distance N events of the 62 2 ” of i 6 L , we obtain the average the boost factor. i . γ ) i /λ i 2 ). L detectors through Pythia 8 − 6.7 e HardQCD:hardbbbar ] we implement a code which deter- − 87 (1 denotes the distance from the IP to the in f.v.] is the individual probability of the · i in eq. ( i i 1 ” and “ N is the speed and /λ L [ i dec N i 1 P β L , with associated Wilson coefficients. We shall N MoEDAL-MAPP − ij – 22 – ) e P generated in the simulation. For all the experiments C , where i 2 N , which indicate the up- and down-type quark genera- , where L ij N c τ in f.v.] = and i i γ i i 1 in f.v.] for the N [ HardQCD:hardccbar β detectors, following ref. [ i L P . For each benchmark scenario, we simulate 10 N , we use formulas for calculating the individual decay probability = b ¯ [ i th sterile neutrino would enter the detector, and P λ ¯ c/b − denotes the average probability of all the simulated sterile neutrinos to c i i MAPP2 → and in f.v.] MoEDAL-MAPP ¯ q, gg q N [ . For scenario 1, the minimal scenario, there is no EFT operator involved, and P MAPP1 5 th sterile neutrino would travel across the detector, if it leaves the detector without h − We emphasize that for the partial decay widths of the heavy mesons into We use the modules “ For the i into light mesons, we take into account the weak interaction via active-sterile neutrino th simulated sterile neutrino to decay in the f.v., discussed in more detail below. − assignment involves two EFT operatorsdimension-6, and of we different drop quark themode flavors. (6) via superscript All the in EFT “production the operatorsconsider following. operator” are two First, ( cases we for fix the thetions production indices considered in a flavor benchmark, respectively. The choices of the flavors should lead To present the resultsconsider of a this subset collider of study possiblesection EFT in operators a and compactwe focus and consider on the representative the full manner three standard we sterile model scenarios charged- described neutrino and in neutral-currents mixing. mediatedspecify via For the their active- scenarios flavor. 2 In and these 3, scenarios which we involve consider EFT 5 operators, different we flavor must assignments. Each for the production and decay processesand of in sterile the neutrinos appendices. are presented in sections 7 Numerical results decay branching ratios. Fromdecay the probability from MC which simulation we with calculate N mixing, the EFT operator(s), and also the interference between them. The computation to simulate the productionprocesses of the charmcorresponding and bottom process, mesons, and respectively, fixmediated including by the both the initial-state the SM mesons weak to interaction and decay the to EFT operators, the with various the channels, corresponding the having decayed. If the traveldetector, direction we of compute the long-lived sterile neutrino points towards the input we use the boostedneutrino, decay denoted length by and the traveling direction of each simulatedmines sterile whether each simulated steriledetector neutrino and travels in if the so directionposition returns pointing towards where the the decay inside the fiducial volume (“f.v.”)i and the total number of sterile neutrinos except with the exponential decay distribution, extracted from existing references [ where JHEP03(2021)148 .  ' D D ρ 2 C | (7.1) (7.2) 4 e and U | B , shown in , -bosons via . ]. The dark 6 . ∓ , Z ∓ , leading to , ) ) e 96 e [ ∓ X X + . To summarize, e for details) and + 11 VRR , which leads to N C ij indicate the flavor 5 (+ (+ + - and ∗ ) ∗ e m e , the and pion D or  ij W π + C + ,K ,D DELPHI , ρ ∓ ∓ 11 ∓ N TRR , m N e e e C K + + → → + ], and   = 4  B D < m π K D 95 : : [ 11 SRR N → → → 21 13 C ) ) m P P N N N JINR C C : : : ( ( ], 11 12 21 ( ) ) ) 94 [ D D D . We present the results in figure C C C the “decay operator” ( 6 ( ( ( 7 – 23 –      PS191 . However, such scenarios probe only a small sterile neutrino . ], ∓ e 5 93 + [  π ), we only include sterile neutrino production via CHARM 7.1 . Here, we lift the requirement of the type-I seesaw relation decaying to N N m and vs.  2 e | 4 + e U N Sterile Neutrino Production Modes : | , and treat the mixing angle and the sterile neutrino mass as two independent free indicates a potential final state meson. See examples below. Second, we simul- Sterile Neutrino Decay Modes : N X As indicated in eq. ( /m It is possible to have only one non-vanishing operator e.g. e 7 ν parameters. The light gray areaincluding shows the the present searches bounds from obtained by various experiments decaying to mass range and insensitivity addition reach require because the of decaying the meson small to production be rate a and vector large particle. decay This width results of in the a vector too mesons. small the active-sterile neutrino mixing.sections Using for the the analytical productionreach expressions and of given the decay in experiments of discussed the inthe the previous section plane sterile neutrino, wem estimate the sensitivity conservative as we underestimate theand number kaon decays of both the via produced the sterile SM weak neutrinos interaction7.1 from as pion well as via The the minimal “decayIn scenario operator” the minimal scenario, the interactions are purely mediated by the following reasons. Despite thesoft larger pions production is not rateslong-lived, well of leading validated these to in mesons, further quick complications. the MCand simulation kaons Finally, simulation are sterile of tools. necessarily neutrinos light, produced Furthermore, resultingwhile the from in we kaons pions limited are show sensitivity reach results in for light sterile neutrinos in the following, these must be taken as 2 and 3 wethrough impose minimal mixing, the see type-I section Seesaw relation todecays, include also weak for interaction lighter contributions masses, sterile respectively. neutrinos We do with not include masses possible production via kaon or pion decays for the Both the production and decay willcoefficients, be which induced we by various discuss operators inand with associated detail decay Wilson below. of sterile Forthe neutrinos all corresponding via scenarios interference we the terms with include SM the the weak EFT production interaction operators. (see For the section theoretical scenarios the decay of sterilecontent neutrinos of via the semi-leptonic final processes. state meson. Here, We the shall consider several decay modes Here taneously turn on another EFT operator to charm and bottom meson decays: JHEP03(2021)148 . e ]. ν 98 , ATLAS ]. This 97 ]. 62 100 ] while the latter 99 between 0.05 eV and exclusion limits, our with those from the e 6 ν experiments, for which m and with a delayed decay, as the 95% C.L. exclusion and takes 3000 signal events µ ANUBIS 3 1 ν MAPP2 − − and 5, to those given in ref. [ . 3 ∼ MAPP1 . These two limits are derived from neutrino N – 24 – /m e ν m ' ], except for the 2 | 4 e 101 U , | are inferior by a factor 6 GeV. For the other experiments, we assume zero background and 85 6 , & 62 N , m investigated a sterile neutrino mixing with 61 , -decays, however with promptly decaying sterile neutrinos. Furthermore, in ] for a related study within the minimal model, with the sterile neutrino pro- W 33 , ATLAS 102 . Results for the minimal scenario with the sterile neutrino mixed solely with the electron 31 ] taking into account an universal efficiency factor 10 experiment considers an integrated luminosity of 3 ab 103 Given the general agreement with the existing results in the literature, we proceed The sensitivity reaches of the various experiments have been determined in the lit- calculation. to evaluate the sensitivitiesthe of sterile these neutrino interactions experiments with toencoded the by a SM EFT particles operators. set are of There enhanced are benchmark by a heavy scenarios, large number new where of physics, possibilities for the flavor structure however only for 100% detector efficiency. HenceComparing we the take sensitivity reach 3 ofliterature, signal the we events experiments find as shown a the inresults good 95% figure shown C.L. agreement in exclusion in figure most limit. difference cases. is For due the to a corrected meson-production-rate and sterile-neutrino-decay-width scenario produced from heavy-flavor mesons rareSee decays ref. has [ not been conductedduced for from ref. [ i.e. a displaced vertex, as well as a promptly decaying sterile neutrino, which mixes with erature [ we present the estimateATLAS for the sensitivity reachbefore for the firstlimit. time. To the Our best estimate of for our the knowledge, a similar estimate for sterile neutrinos in the minimal 0.12 eV with the relation oscillation and cosmological observations,least respectively. one The active formerimposed neutrino finds an mass that upper there eigenstate limit is of of 0.12 at mass eV for at the least sum of 0.05 eV the active [ neutrino masses [ Figure 6 neutrino. gray area corresponds toWe also the show part a excluded brown band by of big “Type-I bang Seesaw target nucleosynthesis region” (BBN) for [ JHEP03(2021)148 we (7.3) (7.4) (7.5) D C and N. P . For scenario C + N,  TRR e + C  + e )0 = 4 ∗ + . In addition, we include ( 0 3 ρ K SRR . C . For the decay operator we ∓ → → e ρ s  for a fixed sterile neutrino mass for a specific flavor choice. We m -mesons, and thus consider the and show the dependence of the + 5 D P − P  C e C D = can reduce the lifetime of the sterile → D D < m N,D N N,D C C + N + ∓ m  e e and the sterile neutrino mass. Both the top and and – 25 – + + P 0 ,  C π ρ ∓ = π → → D + 0  C 1 GeV appears to be disfavored by BBN considerations.  e . . For scenario 2, the leptoquark scenario, for → N vs. the production operator 21 N ) m D P N,D C C + ∓ e N,D + + resulting in the decays   presents the sensitivity reach of all considered experiments for the flavor . e π 11 ) . The following decays then produce sterile neutrinos N 7 D → → m C 0  VLR D C D = 1 GeV. In the bottom row we have fixed Figure We note that in the following EFT flavor benchmarks, with the choice of the canonical N bottom panels show that thecating sensitivity that reach the specific in Lorentz scenarios structureoverall 2 of sensitivity. and the In EFT 3 the interactions are upper-left does ratherhorizontal and not similar, greatly lower-right and part affect indi- of the vertical, the respectively. topduction plots (horizontal, In the upper curves this left) become part or decay of (vertical, the lower parameter right) of space sterile either neutrinos the through pro- benchmark 1. The leftpanels panels to correspond scenario to 3 theoryof (anomalous scenario the vector 2 decay interactions). operator (leptoquark) Inm and the the upper right rowsensitivity we of plot the the experiments on value The essential features of this benchmarkproduction are and summarized decay in modes table viareach minimal in mixing which extends both the upper and lower We are sensitive tochoose sterile ( neutrino masses consider scalar and tensor operators3 that corresponding are to related anomalous through vectorboth interactions, the production and decay operators are 7.2 Flavor benchmark 1 We consider the theoretical scenariosfocus 2 on and sterile 3 neutrinoproduction of production operator section ( through decay of However, the inclusionneutrinos, of possibly the circumventing BBN EFT constrainton operator and the leading EFT to Wilson aparticles, coefficients. potential affecting the Different lower primordial bound flavor helium assignments anddetailed result deuterium study abundances in is to necessary different different to extents. final-state investigateon the A EFT limits operators that can and be we set do from not BBN consideration present BBN exclusion bands below. understanding of theThe general calculations features can easily and befrom the repeated specific sensitivity for UV-complete other reach scenarios. flavor of choices, for various instance experiments. those inspired type-I Seesaw relation, of the EFT operators. Here we consider a few representative flavor choices, to get an JHEP03(2021)148 , vs. 5 D -1 -1 -1 -1 -1 -1 -1 -1 -1 C POT 20 -1 10 :30fb 300 b: 10 × - 2.0 (80) TeV. TA:3ab 3 ATLAS: fb 250 : AL3X AP:3 fb 30 MAPP1: NBS ab 3 ANUBIS: AE2 ab 3 FASER2: AE:10fb 150 FASER: AP:30fb 300 MAPP2: SHIP:2 AHSA ab 3 MATHUSLA: CODEX O -2 10 = D -1 , -1 -1 -1 -1 ) 1.5 P -3 21 11 VLR VLR X 10 ∓ /C AP:3 fb 30 MAPP1: AE2 ab 3 FASER2: AE:10fb 150 FASER: fb 300 MAPP2: C C AHSA ab 3 MATHUSLA: e 2 -mesons decays. (+ 21 VLR [GeV] v C N  + D m -4 -1 e p -1 10 -1 -1 POT  1.0 20 10 + :30fb 300 b: ∼ × - , ρ TA:3ab 3 ATLAS: fb 250 : AL3X NBS ab 3 ANUBIS: Flavor benchmark 1.3 . The color code for the various SHIP:2 N CODEX ∓ -5 10 N e → m + 0.5 s  , and vary the sterile neutrino mass -6 ), there is no sensitivity in the upper vs. , indicating that EFT operators can 10 π D /D 5 P 0 C 10GeV 1.0 = − 21 11 TRR TRR C N → m C C = /D 10 -7 FASER 0.0 10 N -8 -6 -7 -1 -2 -3 -4 -5 -6 -7 -1 -2 -3 -4 -5  P

10 10 10 10 10 10 10 10 10 10 10 10 10 10 10

VLR VLR VLR

= ≤ C C C

11 21 = 4 11 = 4 C D , and P,D – 26 – 21 11 SRR SRR C and shows -1 C C -1 -1 -1 -1 -1 -1 -1 -1 POT D 20 -1 10 :30fb 300 b: 10 C × - MAPP1 Flavor benchmark 1.2 2.0 , TA:3ab 3 ATLAS: fb 250 : AL3X AP:3 fb 30 MAPP1: NBS ab 3 ANUBIS: AE2 ab 3 FASER2: AE:10fb 150 FASER: AP:30fb 300 MAPP2: SHIP:2 = AHSA ab 3 MATHUSLA: CODEX ). This is caused by the rather weak detector acceptance P D -2 P P D D 10 C C C C C C -1 -1 -1 -1 -1 1.5 CODEX-b -3 10 21 TRR AP:3 fb 30 MAPP1: AE2 ab 3 FASER2: AE:10fb 150 FASER: fb 300 MAPP2: AHSA ab 3 MATHUSLA: [GeV] C =4· N m 21 and large SRR -4 -1 C -1 10 -1 -1 POT 1.0 20 P 10 :30fb 300 b: × - C TA:3ab 3 ATLAS: fb 250 : AL3X NBS ab 3 ANUBIS: SHIP:2 CODEX -5 denotes any additional final state particles. 10 decay operator: decay process via X 0.5 -6 . Results for the sensitivity reach for flavor benchmark 1. On the left we have the 10 production operator: . Summary of flavor benchmark 1 for the theoretical scenarios 2 and 3 of section production process via 10GeV 1.0 = N m and jointly the Wilson coefficients. The plots for scenarios 2 and 3 look rather similar For some experiments ( In the lower set of plots, we assume equal -7 0.0 10 -4 -5 -6 -7 -8 -5 -6 -7 -1 -2 -3 -1 -2 -3 -4 , where as the lower case is for

10 10 10 10 10 10 10 10 10 10 10 10 10 10 10

N

SRR TRR SRR TRR TRR SRR = · = · =

C C 4 C C 4 C C 4 ·

= P

21 11 11 21 11 11 left corner (small of these experiments for the light sterile neutrinos producedm from This scale does notthe include EFT possible operators small and dimensionless is couplings thus or only loop indicative suppressions of of the sensitivity range. EFT operators becomes sub-leadinging. with respect This roughly to happens thedominate contributions for over minimal from couplings interactions minimal for a mix- new physics scale of Λ Figure 7 leptoquark-like case, and onC the right the VLRexperiments case. is For shown each in case each the figure. upper figure shows Table 3 respectively. JHEP03(2021)148 . , 5 = − D 21 VLR 10 (7.6) · C . This C MAPP2 2 , sterile 12 , = ) ∼ D D P D , should be at 1.2 GeV. C C C , the boosted > m N = FASER2 AL3X m N P m C with , and at the aforementioned MATHUSLA and on the right P with the same format as C . , and then 8 4 -meson decays via the weak 12 TRR SHiP − ∗ ∓ B vs. ANUBIS C level (corresponding to scales , K 10 , and on the right 3 D · − = 4 + C , we underestimate the production 21 π  TRR e at 3 C . 12 SRR D but for the decay now set ( → C < m MATHUSLA = 4 . 21 , and finally = N N , which reaches roughly ) 5 < m 7 MAPP1 P m D − 21 SRR N C , and to a lesser extent C , sterile neutrinos can still decay leptonically 10 C SHiP π level. m · – 27 – and values can still lead to detectable rates of sterile 4 = goes beyond the kinematical thresholds set by the P − SHiP , P C < m N C ∓ N m K , and m + 8 GeV. For couplings at this level, both the production and .  1 ) for masses e reaches couplings at the 10 AL3X 6 < → N N are large enough, sufficiently many sterile neutrinos are produced. FASER at roughly the 10 D < m C level. The hierarchy in sensitivity reach shown by the various experiments and ATLAS 5 GeV 5 . P − is reduced due to the need to produce a heavier kaon in the final state. For the EFT operators no longer contribute to the decay rate and the SM weak -meson masses. For C 10 . On the left we consider D N · K , and 7 2 summarizes the details of this scenario. m 4 . In the upper row, we show plots in the plane ≤ < m The hierarchy in sensitivity of the different experiments is also very similar. In the The sensitivity limits for this scenario are shown in figure Again, the sensitivity reach in N 12 VLR P,D lower panels we seelighter some differences compared tom flavor benchmark 1.interaction The becomes sensitivity the only to mechanism for sterile neutrinos to decay and be detected. This in figure Similarly for the decayC we have on theSimilar left features as in thesensitivity previous in scenario the are event observed rates and to there the seems specific hence to final-state be meson. little leads to sterile neutrino decay processes Table right-hand side of the two lower plots of figure 7.3 Flavor benchmark 2 In flavor benchmark 2 we chooseFor a the different flavor-structure production for the operator decay we Wilson coefficient. take again ( neutrinos for this benchmark cancurrent. still If be produced fromFurthermore, the specifically for decay lengths of theseranges. sterile This neutrinos corresponds can fall to into the the extended respective sensitive geometric parameter sensitivity regions, as shown on the pion and via the weak interaction.neutrino Thus production. larger We stress againof that sterile for neutrinos by omitting production via pions and kaons. For of roughly 8 TeV inCODEX-b Λ). Next in sensitivitysensitive is to couplings downC to around 5 is essentially the sameminimal in scenario (see scenarios figure 2 and 3, and is very similar to the hierarchy in the most sensitive experiment would clearly be in the range 0 decay of sterile neutrinosplays are a still sub-leading dominated role. byunder the consideration. The EFT operators sensitivity and then minimal depends mixing a lot on the experimental setup although in scenario 2, the sensitivity to smaller sterile neutrino masses is a bit better. The JHEP03(2021)148 D C - (7.7) (7.8) P -1 -1 -1 -1 -1 -1 -1 -1 -1 POT C 20 -1 10 :30fb 300 b: 10 × - . 2.0 TA:3ab 3 ATLAS: fb 250 : AL3X AP:3 fb 30 MAPP1: NBS ab 3 ANUBIS: . The two AE2 ab 3 FASER2: AE:10fb 150 FASER: AP:30fb 300 MAPP2: SHIP:2 AHSA ab 3 MATHUSLA: CODEX 7 9 N. -2 10 + -1 N, -1 -1 -1 -1 ∓ ) 1.5 e -3 + ∓ 21 12 VLR VLR X 10 e + AP:3 fb 30 MAPP1:  AE2 ab 3 FASER2: AE:10fb 150 FASER: fb 300 MAPP2: C C AHSA ab 3 MATHUSLA: e (+ 21 VLR [GeV] +  C ) N  ∗ m + -4 -1 e ( -1 ∗ 10 -1 -1 POT 0 1.0 20 ρ K 10 + :30fb 300 b: × - TA:3ab 3 ATLAS: fb 250 : AL3X ,K NBS ab 3 ANUBIS: Flavor benchmark 2.3 SHIP:2 N CODEX → → -5 ∓ (right) scenarios in the 10 s e  → + 0.5 s VLR -6  10 C /D K 0 12GeV 1.2 = 21 12 TRR TRR N . m C C → /D -7 N,B 5 N,B 0.0 10 -8 -6 -7 -1 -2 -3 -4 -5 -6 -7 -1 -2 -3 -4 -5 

10 10 10 10 10 10 10 10 10 10 10 10 10 10 10

VLR VLR N VLR

= + C C

C + 12 21 = 4 12 = 4 D ∓  e e . This leads to the sterile neutrino production – 28 – 21 12 SRR SRR + + 13 P -1 C C -1 -1 -1 -1 -1 0 -1 -1 -1  POT C 20 -1 π ρ 10 :30fb 300 b: 10 × - Flavor benchmark 2.2 2.0 TA:3ab 3 ATLAS: fb 250 : AL3X AP:3 fb 30 MAPP1: NBS ab 3 ANUBIS: AE2 ab 3 FASER2: AE:10fb 150 FASER: AP:30fb 300 MAPP2: SHIP:2 → → AHSA ab 3 MATHUSLA: CODEX 0  . Summary of flavor benchmark 2. -2 P P D D 10 C C C C -1 -1 -1 -1 -1 1.5 -3 10 21 TRR Table 4 AP:3 fb 30 MAPP1: AE2 ab 3 FASER2: AE:10fb 150 FASER: fb 300 MAPP2: AHSA ab 3 MATHUSLA: N,B [GeV] C =4· N m 21 SRR -4 -1 + C -1 10 -1 -1 POT 1.0 20 ∓ 10 :30fb 300 b: × - e N,B TA:3ab 3 ATLAS: fb 250 : AL3X NBS ab 3 ANUBIS: . Results for flavor benchmark 2. The format is the same as in figure SHIP:2 CODEX + -5 + 10 decay operator:   decay process via e π 0.5 -6 → → 10 production operator: 0 Figure 8 production process via  12GeV 1.2 = N B B m -mesons. Compared to flavor benchmark 1, we keep the same flavor structure for the Our results for the sensitivity reach for this benchmark are shown in figure -7 0.0 10 -4 -5 -6 -7 -8 -5 -6 -7 -1 -2 -3 -1 -2 -3 -4 B

10 10 10 10 10 10 10 10 10 10 10 10 10 10 10

SRR TRR SRR TRR TRR SRR

= · = · = C C 4 C C 4 C C 4 · =

21 12 12 21 12 12 top panels show results for the leptoquark (left) and The relevant information is summarized in table 7.4 Flavor benchmarkWe 3 proceed to study a scenarioof where sterile neutrinos are mainlydecay produced through Wilson decays coefficient, butvia turn the on decay processes leads to a furtherare reduction conservative in as sensitivity. we We have stress not again considered sterile that neutrino our production results via in kaon this decays. regime JHEP03(2021)148 5.5 . -1 -1 7 . The -1 -1 -1 -1 -1 -1 -1 5.0 POT 20 -1 10 :30fb 300 b: 10 -mesons × - TA:3ab 3 ATLAS: fb 250 : AL3X 4.5 AP:3 fb 30 MAPP1: B NBS ab 3 ANUBIS: AE2 ab 3 FASER2: AE:10fb 150 FASER: AP:30fb 300 MAPP2: SHIP:2 AHSA ab 3 MATHUSLA: AL3X CODEX showed the -2 4.0 10 -1 -1 , and -1 -1 collisions and by -1 SHiP 3.5 ) -3 13 11 VLR VLR 10 X pp ∓ 3.0 AP:3 fb 30 MAPP1: AE2 ab 3 FASER2: AE:10fb 150 FASER: fb 300 MAPP2: C C AHSA ab 3 MATHUSLA: e 13 VLR (+ [GeV] C N +  ANUBIS m 2.5 -4 -1 e -1 10 -1 -1 POT ,  20 10 :30fb 300 b: + × - 2.0 , ρ TA:3ab 3 ATLAS: fb 250 : AL3X NBS ab 3 ANUBIS: Flavor benchmark 3.3 SHIP:2 CODEX N ∓ -5 -mesons produced and that of collision experiments. Second, 10 e 1.5 B − → and show sensitivity curves as a + 20 at 14 TeV s pp 1.0 MATHUSLA  D -6 ∼ 10 π /B C 0 0.5 26GeV 2.6 = 13 11 TRR TRR N → = m C C /B -7 0.0 P 10 N -8 -6 -7 -1 -2 -3 -4 -5 -6 -7 -1 -2 -3 -4 -5 

10 10 10 10 10 10 10 10 10 10 10 10 10 10 10

VLR VLR VLR

= C C C C

11 13 = 4 11 = 4 B is sufficiently large to ensure sterile neutrinos D 5.5 -meson decays. In the earlier flavor benchmarks, – 29 – 13 11 SRR SRR C D -1 C C -1 -1 -1 -1 -1 -1 -1 -1 5.0 POT 20 -1 10 assume :30fb 300 b: 10 × - Flavor benchmark 3.2 than at the 14 TeV TA:3ab 3 ATLAS: fb 250 : AL3X 4.5 9 AP:3 fb 30 MAPP1: NBS ab 3 ANUBIS: AE2 ab 3 FASER2: AE:10fb 150 FASER: AP:30fb 300 MAPP2: SHIP:2 AHSA ab 3 MATHUSLA: CODEX . Summary of flavor benchmark 3. -2 P P D D 4.0 10 C C C C SHiP -1 -1 -1 -1 -1 3.5 no detection is possible in any of the experiments, even for large -3 -meson by roughly a factor 7 10 13 TRR Table 5 3.0 D AP:3 fb 30 MAPP1: AE2 ab 3 FASER2: − AE:10fb 150 FASER: fb 300 MAPP2: AHSA ab 3 MATHUSLA: . 6 GeV. The resulting curves are rather different from the scenarios, [GeV] . C =4· 10 N m 13 2.5 SRR -4 -1 C -1 10 -1 -1 POT < = 2 20 SHiP 10 :30fb 300 b: × - P 2.0 N TA:3ab 3 ATLAS: fb 250 : AL3X NBS ab 3 ANUBIS: C SHIP:2 CODEX -5 m 10 1.5 decay operator: decay process via . Results for flavor benchmark 3. The scenarios and the labeling are as in figure 3000 at 1.0 -6 10 production operator: ∼ production process via 0.5 26GeV 2.6 = N m The two lower panels in figure -7 can be turned off and sufficient sterile neutrinos will be produced via minimal mixing 0.0 . This lack of sensitivity is explained by the fact that the production rates of 10 -4 -5 -6 -7 -8 -5 -6 -7 -1 -2 -3 -1 -2 -3 -4

Figure 9

10 10 10 10 10 10 10 10 10 10 10 10 10 10 10

-mesons is much smaller at

SRR TRR SRR TRR TRR SRR

= · = · = D P C C 4 C C 4 C C 4 · =

13 11 11 13 11 11 D a factor function of the sterilestrongest neutrino mass. sensitivity, but In here thereason it previous is flavor performs twofold. benchmarks worse First, than the ratio between the number of to still detect sterile neutrinos,decay as in long the as respective detectorscenario and volumes. for This feature hasC disappeared in this benchmark are smaller than that of plane for fixed where sterile neutrinos are producedC via JHEP03(2021)148 5.5 -1 . -1 -1 -1 -1 -1 -1 -1 -1 5.0 POT 20 -1 7 10 :30fb 300 b: 10 × - TA:3ab 3 ATLAS: fb 250 : AL3X 4.5 AP:3 fb 30 MAPP1: NBS ab 3 ANUBIS: AE2 ab 3 FASER2: AE:10fb 150 FASER: AP:30fb 300 MAPP2: SHIP:2 AHSA ab 3 MATHUSLA: CODEX -2 4.0 . The relevant 10 12 D -1 -1 -1 -1 -1 ) 3.5 C -3 13 12 VLR VLR X 10 C C 3.0 AP:3 fb 30 MAPP1: AE2 ab 3 FASER2: AE:10fb 150 FASER: fb 300 MAPP2: AHSA ab 3 MATHUSLA: (+ 13 VLR [GeV] C ∓ N  m 2.5 e -4 -meson decay, as for the e -1 -1 10 -1 -1 POT 20 B + + 10 :30fb 300 b: × - 2.0  Flavor benchmark 4.3 TA:3ab 3 ATLAS: fb 250 : AL3X NBS ab 3 ANUBIS: ) N SHIP:2 CODEX -5 ∗ 10 ( 1.5 → K s 1.0 -6 → 10 /B 0 0.5 28GeV 2.8 = N 13 12 TRR TRR N 8 GeV. In general these plots show very m C C . /B -7 , except for an overall small reduction in sensitivity to couplings at the impressive 0.0 10 -6 -7 -8 -5 -6 -7 -1 -2 -3 -4 -5 -1 -2 -3 -4 

10 10 10 10 10 10 10 10 10 10 10 10 10 10 10

VLR VLR

VLR 9 = C C C

12 13 = 4 12 = 4 B detector for the long-lived sterile neutrinos. = 2 N 5.5 (100) TeV. – 30 – 13 12 SRR SRR m O -1 SHiP C C -1 -1 -1 -1 -1 -1 -1 -1 5.0 POT 20 -1 MATHUSLA 10 :30fb 300 b: 10 × - . Flavor benchmark 4.2 TA:3ab 3 ATLAS: fb 250 : AL3X 4.5 AP:3 fb 30 MAPP1: NBS ab 3 ANUBIS: AE2 ab 3 FASER2: AE:10fb 150 FASER: AP:30fb 300 MAPP2: SHIP:2 6 AHSA ab 3 MATHUSLA: CODEX -mesons, are less boosted in the very forward direction, -2 . Summary of flavor benchmark 4. 4.0 P P D D 10 B C C C C -1 -1 -1 -1 -1 3.5 -3 10 plane for fixed 13 TRR 3.0 AP:3 fb 30 MAPP1: AE2 ab 3 FASER2: Table 6 AE:10fb 150 FASER: fb 300 MAPP2: AHSA ab 3 MATHUSLA: [GeV] D C =4· N C m 13 2.5 SRR -4 -1 - C -1 10 -1 -1 POT P 20 10 :30fb 300 b: × - C 2.0 TA:3ab 3 ATLAS: fb 250 : AL3X NBS ab 3 ANUBIS: SHIP:2 CODEX -5 10 1.5 . Results for flavor benchmark 4. The plot format is the same as in figure shows the numerical results for this benchmark. In the two top panels we decay operator: decay process via 1.0 10 -6 10 production operator: level, corresponding to scales of 0.5 28GeV 2.8 = production process via N 6 m − -7 Figure 10 Figure 0.0 10 -7 -3 -4 -5 -6 -7 -8 -3 -4 -5 -6 -1 -2 -1 -2

10 10 10 10 10 10 10 10 10 10 10 10 10 10 10

10

SRR TRR SRR TRR TRR SRR

· = = · = C 4 C C 4 C C 4 C · =

13 12 12 13 12 12 · For flavor benchmark 4 theprevious benchmark, production but primarily here proceeds sterileinformation via neutrinos is decay summarized to in a table kaon through show plots in the similar features as their counterparts in figure scenario and the other flavorprevious benchmarks. flavor However, benchmarks the overall with5 reach the is increased over the 7.5 Flavor benchmark 4 given their larger masses,leading the to weakened acceptanceThe of hierarchies the in sensitivity of the other experiments are the same as in the minimal JHEP03(2021)148 5.5 -1 -1 -1 -1 -1 -1 -1 -1 -1 5.0 POT 20 -1 . 10 :30fb 300 b: 10 × - 7 TA:3ab 3 ATLAS: fb 250 : AL3X 4.5 AP:3 fb 30 MAPP1: NBS ab 3 ANUBIS: AE2 ab 3 FASER2: AE:10fb 150 FASER: AP:30fb 300 MAPP2: SHIP:2 AHSA ab 3 MATHUSLA: CODEX -2 4.0 10 . However, for K -1 -1 ) -1 -1 ) -1 3.5 m -3 21 13 VLR VLR X . Compared to the X 10 C C 3.0 N AP:3 fb 30 MAPP1: AE2 ab 3 FASER2: AE:10fb 150 FASER: fb 300 MAPP2: & AHSA ab 3 MATHUSLA: (+ (+ 13 VLR [GeV] m C  ∓ N  N m e 2.5 e -4 e -1 -1 10 -1 -1 POT . In this case, the decay m 20 vs. + + + 10 :30fb 300 b: × - 21 D 2.0  D Flavor benchmark 5.3 TA:3ab 3 ATLAS: fb 250 : AL3X NBS ab 3 ANUBIS: N C ) N SHIP:2 CODEX -5 ∗ C 10 ( 1.5 → → D = s s and 1.0 P -6 → 10 /D /B C 13 P 0 0 0.5 35GeV 3.5 = N C 21 13 TRR TRR N m C C /B /D -7 0.0 10 -6 -7 -8 -5 -6 -7 -1 -2 -3 -4 -5 -1 -2 -3 -4  

shows the resulting sensitivity reach. In

10 10 10 10 10 10 10 10 10 10 10 10 10 10 10

VLR VLR VLR

=

C C

= 4 C = 4 21 13 B 21 D 11 5.5 – 31 – 21 13 SRR SRR -1 C C -1 -1 -1 -1 -1 -1 -1 -1 5.0 POT 20 -1 10 :30fb 300 b: 10 × - Flavor benchmark 5.2 TA:3ab 3 ATLAS: fb 250 : AL3X 4.5 AP:3 fb 30 MAPP1: NBS ab 3 ANUBIS: AE2 ab 3 FASER2: AE:10fb 150 FASER: AP:30fb 300 MAPP2: SHIP:2 . Figure AHSA ab 3 MATHUSLA: CODEX 7 -2 . Summary of flavor benchmark 5. P P 4.0 D D D 10 C C C C C -1 -1 -1 -1 -1 3.5 -3 10 13 TRR 3.0 AP:3 fb 30 MAPP1: AE2 ab 3 FASER2: Table 7 AE:10fb 150 FASER: fb 300 MAPP2: AHSA ab 3 MATHUSLA: , they show similar exclusion limits for [GeV] C =4· N m 13 2.5 9 SRR -4 -1 C -1 10 -1 -1 POT 20 10 :30fb 300 b: × - 2.0 TA:3ab 3 ATLAS: fb 250 : AL3X NBS ab 3 ANUBIS: . Results for flavor benchmark 5. The format is the same as in figure SHIP:2 CODEX because of the choice for a slightly larger mass of the sterile neutrino. The -5 10 1.5 P decay operator: C decay process via 1.0 -6 10 production operator: 0.5 35GeV 3.5 = production process via production process via Figure 11 N m -7 0.0 10 -7 -3 -4 -5 -6 -7 -8 -3 -4 -5 -6 -1 -2 -1 -2

10 10 10 10 10 10 10 10 10 10 10 10 10 10 10

SRR TRR SRR TRR TRR SRR

· = = · = C 4 C C 4 C C 4 C · =

13 21 21 13 21 21 7.6 Flavor benchmark 5 In flavor benchmark 5,operator also we leads turn to production onrestricted of the to sterile neutrinos, operators a but mass the rangethe resulting where sterile benchmark they neutrinos features are can only in decay table via minimal mixing. We summarize lower panels of figure lighter sterile neutrinos, since onlyneutrino the decay mixing modes are via open, thevarious weak the experiments current is sensitivity and the active-sterile is same significantly as reduced. in the The previous hierarchies benchmark of for the both EFT scenarios. the reach of two lower plots show sensitivity curves in the plane JHEP03(2021)148 . 6 up − NS), than 10 ν · . More τ 5 FASER m SHiP ∼ are largely − T B decay, consid- E

β + < m leptons, but there l 8 TeV for τ N ∼ . Assuming a scaling of < m mesons in the final states. ), while experiments such π τ m (6) QuνL . In this case sterile neutrinos + C D for which the sensitivity grows τ is sensitive to Wilson coefficient SMEFT operators on lepton fla- FASER2 ν m < m for -mesons is much larger at , and 4 SHiP N B − FASER m -mesons, and the sterile neutrino can, in 10 (6) SMEFT Lagrangian. In the mass range LdQν -decays are examined. The most stringent B ν C ∼ - and β , D - or – 32 – D (6) LνQd ), we see that such limits can be used to constrain 36 TeV. However, decays allow for a neutrino can reach down to coupling strengths of C , , the sensitivities range from Λ & 2.11 are comparable to the previous flavor benchmark, but 2 Λ (6) D duνe SHiP / 2 C C v , becomes the most sensitive experiment in fact, because the ∼ , and and (this is extended to (6) Hνe P 3 C − C SHiP AL3X ) and the particular flavor configuration, as long as the sterile neu- 10 , the sensitivity curves are rather stable with respect to different EFT ] consider a larger set of decays corresponding to ∼ ] investigated limits from pion decays, tau decays, and singular leptons 2.18 more comparable to our studies, while searches for B 107 , where 105 N , , ANUBIS D m , < m 5 TeV. We did not explicitly consider processes involving 106 104 SMEFT operators we consider here can also be probed in other experiments. N − ν < m 2 100 TeV for the larger experiments. N SMEFT operators decays and an efficiency factor for reconstructing decays of & ∼ < m Refs. [ Refs. [ The ν m τ MATHUSLA K quantitative statements require a detailedvia study that includes sterile neutrino production several flavor configurations. Additionally, thevor effects universality of (LFU), CKM unitarity, and with missing transverse energy.obtained The most from restrictive pion bounds decaysmass on with the range new Λ physicsindependent scale of are sterile neutrino masses.to Λ The latter investigations setshould be the good new sensitivity physics in scale the appropriate mass range work. For instance, limitssterile from neutrino pion masses decay below orerably the from lower pion than or mass the nuclear or GeV-scalegive beta the sterile an decay respective neutrinos overview require Q of considered value the in of literature. this work. Here we briefly to Λ These include meson and tau decays,missing elastic coherent transverse neutrino- energy scattering (EC searches, etc.trino Depending mass on range the and probe, the the relevant flavor sterile assignment neu- can differ from the cases considered in this couplings of about as From the matching relations inthe eq. ( these Wilson coefficients as m operators in eq. ( trino can be producedturn, via decay the semi-leptonically. decay of assignment While requires each a particular detailed choice study, of we find EFT that operators and flavor at the other experiments. 8 Discussion and comparisonOur with results other indicate probes that proposedsitive experiments to to higher-dimensional detect long-lived operators particles in are very the sen- relative sensitivities to the sensitivity in the bottomonly panel drops decay a via bit for minimalfor mixing. Theproduction exception cross section is difference between the upper row we fix the sterile neutrino mass at 3.5 GeV. In general the absolute and JHEP03(2021)148 ]. = 1 108 ∗ 1 TeV. RL SMEFT 31 & overtakes y ν decay rates LR 11 ]. For flavor y 100 MeV) [ = νββ 113 , rates have a very ∗ . ] are considered in . We stress that the 51 MATHUSLA RL 11 N y m νββ 109 12 LR 11 (0 y LQ are somewhat stronger than m decay rates and determine the involving an , a down νββ = 1 and ] some of us developed a framework and experiment [ νββ ∗ 51 N (6) QuνL RL 11 , chosen as representative experiments, m C y LR 21 y , and COHERENT = decay experiments are only valid for Majorana ∗ – 33 – MATHUSLA 8 TeV. Similar sensitivities are found examining (6) LdQν RL 11 − y C νββ 5 and ), the limits from 0 . , including light sterile neutrinos ( LR 11 0 y ν 12 & experiments. In ref. [ (6) LνQd cτ C ]. FASER2 → experiments, which only depend on sterile neutrino couplings νββ b NS based on the 112 5 MeV) and the resulting bounds are at the level of Λ ν , . νββ 0 56 100 MeV, which drops off for larger or smaller masses. To make a . ' decay rates in the presence of light sterile neutrinos and the ]. Sterile neutrinos considered in these works are again much lighter than N N sensitivity curves as a function of m 111 m limits for masses between 1 and 5 GeV. νββ experiments are sensitive to a broad range of neutrino masses with a peak , limits albeit in a much small sterile neutrino mass range. SHiP 110 νββ , νββ 74 (110) TeV in the limit of massless sterile neutrinos and thus cannot be directly νββ & 106 We stress that the bounds from 0 For these choices of couplings, we can calculate 0 In this work we have focused on Majorana neutrinos (although our sensitivity curves electrons, and in generalthan to 0 a broader range of quark flavors. They are thus more general in the relevant masscurrent range. 0 For flavor benchmark 3 theneutrinos prospected and final-state electrons.experiments discussed However, here the are sensitivityappropriate curves mass (roughly) range for valid also the for for various (pseudo-)Dirac couplings LHC neutrinos, to and muons and, in to the lesser extent, taus instead of this dependence here for simplicity).uncertainties The associated results with are shown hadronic inare and figure sizable nuclear and matrix not elementsbenchmark for included 1 0 in (left the panel plot,the of prospected for figure sensitivity details of we refer to refs. [ 3 to the sensitivity ofto first-generation 0 quarks and leptons. LHC and small dependence on phases appearing in the 3 + 1 neutrino mixing matrix and we neglect to probe scenario 2:states, 0 the 3 +sensitivity 1 at leptoquark model.comparison, As we sterile consider neutrinos theand appear case vanishing as couplings virtual for other flavors. We can then compare flavor benchmarks 1 and are not affected dramatically if weconstraints had can considered be Dirac set from neutrinos 0 instead)to for which calculate strong 0 Lagrangian. In particular, we investigated the reach of current and future experiments Further, limits from EC refs. [ the GeV-scale ( We conclude that the sensitivitiesand of the complementary experiments to considered existing herediscussed are constraints. e.g. competitive in with Constraints refs. on [ dimension-five couplings are (strange) quark and anby electron Λ using LFU constraints.compared The to new results physics obtainedbinations scale here. are is in Bounds limited the onanomalies range in other the of operators transition Λ and different flavor com- bounds are on the operators JHEP03(2021)148 Xe 5 136 4 3 [GeV] N m (yellow) for the leptoquark 2 . A large number of mesons are to detect sterile neutrinos. 1 MATHUSLA LHC collaborations, various proposed SHiP 80 60 40 20

100

CMS

MATHUSLA LQ ] [ TeV m (red) and and 2.0 and – 34 – and a number of so-called far detectors at various FASER2 SMEFT). This framework allows for a light gauge- ν ATLAS SHiP FASER 1.5 , the SM Higgs vacuum expectation value. This framework de- [GeV] v N 1.0 m  M 0.5 . Comparison between constraints from neutrinoless double beta decay of ] and projected sensitivity of 114 To avoid theoretical bias we approached this problem in the framework of the neutrino- In this work, we focused on relatively light sterile neutrinos with masses at the GeV

40 30 20 10 60 50

-collision interaction points e.g. LQ ] [ TeV m extended SM effective fieldsinglet theory fermion, ( a sterileto neutrino have or heavy masses neutralscribes lepton, effective and local assumes interactions otherexpansion. between BSM sterile fields We neutrinos considered and SM dimension-6 fields operators in that a allow systematic for sterile neutrino produc- pp expected to be produceddecay at the to interaction sterile points neutrinos. ofbottom these and We experiments, focused which charm in on mesonreach turn sterile of decays can present neutrinos in and which hadronic future can LHC collisions, be experiments, and and produced investigated from the sensitivity leptogenesis and opens up thethe possibility decays of of efficiently pseudoscalar producingIn mesons, sterile particular which neutrinos are at through copiously CERN,experiments produced besides in the are collider presently experiments. undersuch as discussion the specifically fixed-target experiment targeting the detection of LLPs, particles through minimal mixing withnew active fields neutrinos, with to masses more well exotic aboveor the scenarios grand electroweak involving unified scale such theories. as left-right symmetric models scale, down to about 100 MeV. This mass range is interesting, as it is linked to low-scale The possibility of sterilefor neutrinos long-lived particles provides one (LLPs).problems in of particle Sterile the physics neutrinos and mainasymmetry cosmology, provide of such motivations the as compelling for Universe. active solutions neutrino the Sterilemodels, masses neutrinos to search are ranging and in major from the fact the predicted minimal in scenario a where variety of they theoretical interact with Standard Model (SM) scenario. In the left(3). (right) panel See we text turned for on more LQ details. couplings corresponding to flavor benchmark9 1 Conclusions Figure 12 (blue) [ JHEP03(2021)148 - D , and MATHUSLA level. The 6 . For the . Of all the 7 3 − and level, for most – − and 25 TeV for 10 3 5 · − MoEDAL-MAPP SHiP , FASER -meson production rates B , cf. tables ANUBIS scaling, ij , 2 M Λ / + 2 AL3X e v , have the most extensive sensitivity → SMEFT operators are generated. The N ν , because of the contributions from SM D is operational. Thus we strongly encourage m MATHUSLA MATHUSLA , – 35 – and ATLAS FASER -meson decays into sterile neutrinos, and the sterile , B SHiP collaborations to perform a proper full analysis of the - or does not show the best performance. Instead, we found that CMS D CODEX-b , SHiP and would be sensitive to Wilson coefficients at the 5 ATLAS study we applied an overall flat efficiency factor of 10 ATLAS -meson decays and sterile neutrinos can decay semi-leptonically. ANUBIS B -meson benchmarks, because of its much smaller experiments for the first time. For the EFT scenarios we consider active- ATLAS B - or and D , which are much smaller experiments. , where Λ is the high-energy scale where the 2 . For the minimal scenario, the obtained sensitivity curves are in agreement with ex- We compared our projected sensitivities with (projected) constraints obtained from For our For our We performed Monte-Carlo simulations to evaluate the sensitivity reach of the con- Λ / 2 a much firmer footing. other probes of sterileson neutrinos with decays, effective tau interactions decays,transverse such as energy, coherent light and neutrino-nucleus pseudoscalar neutrinoless scattering, me- double LHC beta searches decay. for Our missing results are very competitive flavor configuration of thethrough EFT operators, as long as sterileexperiments neutrinos we can have studied be hereour produced only colleagues at scenarios we have presented and investigated here. This should put our approximations on FASER2 and a weaker acceptance, MATHUSLA sensitivity curves appear to be fairly independent of the Lorentz structure and the exact v sensitivity drops at the edges ofeven the for allowed mass sterile range, but neutrinosweak does interactions with not and disappear masses active-sterile completely, mixing. above could Assuming probe a scales around 80 TeV. This scale is lowered to 8 TeV for reach. They are sensitiveof to the dimensionless kinematically allowed Wilson sterileproduction coefficients neutrino at and mass the decay range. 10 of Fortors sterile such with minimal values neutrinos sterile-active of is mixing couplings, playing dominatedcouplings the a subleading by and role. the potential Apart higher-dimensional loop from dimensionless opera- suppressions, the dimensionless Wilson coefficients scale as sterile neutrino mixing, theFor EFT each operators, EFT scenario, andEFT we their operators considered interference induce a terms either seriesneutrinos simultaneously. decay of to different an electron flavor andmeson benchmarks, various benchmarks, mesons, where we the found that where higher-dimensional operators are turned off. sidered experiments: SHiP isting results with minimal discrepancies,MoEDAL-MAPP while we obtained sensitivity curves for the two considered specific scenarios whereat a the subset, same or time. onlyleft-right-symmetric a models. These single, Other scenarios EFT EFTstraightforwardly are operator scenarios investigated or motivated is by specific by turned using UV-complete UV on thebenchmark models completions extensive our can like formulae calculations be given leptoquark with in or the the literature, appendix. we also To considered the minimal scenarios tion via mesonic decays at tree level. Our framework is general, but for concreteness we JHEP03(2021)148 (A.3) (A.1) (A.2) mesons. Furthermore π , ’s. This will restrict the ) τ S µ M  and ν if q K , ≡ SMEFT operators and of sterile , − ν µ M . Current-algebra or leading-order ν M  q f  ∗ f µ µ j M q q iq ( m m T M 2 M V M f + i ≡ m ) i if q q ( m i i ≡ i ≡ − M ) ) | – 36 – j = q i 5 q,  q,  ) ( ( γ q ∗ ∗ µ ( γ M M i | | M ¯ q | j j | j q q are an excellent probe of 0 q h µ 5 µν γ γ i σ i ¯ q i ¯ q | ¯ q | SHiP | 0 0 h 0 h h , we define meson decay constants via j q is a pseudoscalar meson with momentum i ) q ( M | A straightforward extension of our work is to final state muons instead of electrons. and a quark i ¯ for the pseudoscalar current.states for The vector vector mesons. and We tensor define currents only induce two-body final where chiral perturbation theory gives Sterile neutrinos can decayteraction into or a EFT charged operators.for meson pseudoscalar and Pseudoscalar and a mesons axial-vector currents, can chargedfinal while state lepton decay vector via into mesons by vector can a or theq decay tensor two-body weak into currents. final a in- For two-body state pseudoscalar mesons containing an anti-quark gram of the Asia Pacificthrough Center BMBF for grant Theoretical 05H18PDCA1. Physics. The work of H. K. D.A is supported Two-body sterile neutrinoA.1 production and decay Charged processes currents by the RHIC Physicssupported Fellow partly Program by of the Ministry thenumber of RIKEN MoST-109-2811-M-007-509, Science BNL and and partly Research Technology (MoST) byPlanning Center. of the of Taiwan Z. Ministry with Korea, grant of S. the Science,Government W. Pohang through ICT is the City & Young Government, Scientist Future Training and Asia the Pacific Economic Gyeongsangbuk-do Cooperation Provincial pro- processes to investigate the sensitivity of various experiments toAcknowledgments heavier sterile neutrinos. We thank Philip Bechtle, Haolin Li, and Vasiliki Mitsou for discussions. JdV is supported production of sterile neutrinosin from the a decay future ofkinematically project, the viable light we regions, but shall wouldefficiencies. consider also require the a Finally, case properthrough investigation in of of parton this final the detection collisions work state whichsterile we neutrinos are have masses subleading below, neglected roughly, with direct 5 GeV. respect sterile It to would neutrino be rare production interesting mesonic to include decays partonic for neutrinos at the LHC and neutrinos in general. Here we expect basicallyneutrino mass the we have same considered. results, A more except involved extension at would be the to also very consider lowest the end of the sterile and complementary. We conclude that searches for displaced vertices of long-lived sterile JHEP03(2021)148 → l , ν (A.4) (A.6) (A.5) ) pro-  2 N V S, f , i ∗ ) 1 ab,i g and polarization 1 → VV, q (6) SRR f C ab,i V S,  2 f 2 f

−

2 i | ∗ VV, ) ) f k , (6) (6) SLR VRR l i 2 ∗ C C ) (6) ) + SRR )( 2 N − C , momentum N , m ∗ SS (6) − VRR ) (6) f VRR M − → C . (6) 2 N VLR . the charged lepton generation, and 2 C ) 2 k

m C (6) , and the corresponding decay − m SLR and we define the functions k 2 N

l M − ab,i m ( C ν ( f − 2 from initial-spin averaging. We then m ( (6) + SRR ). We begin with neutrino production , / 2 k − + (6) 2 − VLR C ijkl + |M

(6) VLR 2 N ) m C (6) , − VLL k 2 k . All decay constants are given below in = − ( l C 2.18 2 N m C ( V + N m M (6) VLL X h h spins f m with mass − ab,i → m C (6) VLL − (6) 2 M SLR g

∗ k 2 k – 37 – + = Re Re C  ' − C ij m 2 M

h m 2 k m ij ij ( M 2 | 2 2 M M S S m ) T M M 2 M N m M ) − f ( ( k f f Re f , and l k S m m M  } 2 2 M 2 M M M M 4 2 f m 2 2 F + f f f 2 , m m 2 G + +( + + 1 M { ≡ ≡ − ≡ ≡ − ≡ = 1 2 1 2 = → 2 SS i | ) f . V S, V S, , VV, VV, N l − ( k f f f f } ν l + |M denotes the generation of quark flavors that make up the meson (we N X spins ij denotes a vector meson → 1 2 i ) the neutrino mass eigenstate. For the decay of the neutrino mass eigenstate − V V, SS, V S × } { M . q,  ¯ 2 n ( ( where ∗ = C − k |M M l ab | ,..., 1 + For sterile neutrino decay, we include the charge-conjugated final states leading to an We obtain for the summed-over-spins squared amplitudes for sterile neutrino ( Armed with these decay constants, we calculate the production and decay rates of { X spins + ij . Heavy-quark symmetry relates = µ where additional factor 2 compensatingobtain the additional 1 where all Wilson coefficients carry flavor indices duction M drop these indices below forl notational convenience), we also include the decaythe to Majorana the nature charge-conjugate of final state which is equally likely due to  appendix neutrino mass eigenstates startingvia from the decay eq. of ( pseudoscalar mesons where JHEP03(2021)148 . (A.9) (A.7) (A.8) 1 (A.12) (A.10) (A.11) ∗ V T,i , ∗ VV, f f  , 2  = 2 2

∗ ab,i . ∗ VT, g ∗ VV, 2 3 f ∗ V T,i f → , i g (6) VRR m , i ∗ 2 2 , ∗ ∗ C ab,i  )  f m 2

∗ (6) TRR TT + ) 2 2 1 ··· f | , C ) 2 N (6) ) VRR 2 − − k + ) ∗ l VT, C m 2 3 (6) VLR f 2 N ,µ = C + c m (6) + i − VRR

A η m 2 1 ∗ j 2 k 2 C ∗ TT N + m 2 − g m (6) 1 2 TRR + (6) 2 VLR 2 k , √

→ C 2( |M| C ∗ − TT m ∗ ( + f ( 2 2 − (6) ∗ VLR is the mass of the decaying particle, (6) V V,i VLL 2 (6) VLL M ) ,

,µ f X m − spins C . ( 1 C A (6) 0 VLL 2 1 ) C ( 2 N

) − j ) 1 n h h C  m 2 N m 6 2 N m N 2 h ) (6) TRR |M 2 1 = 2 2 3 m ) √ m Re Re m C + ∗

∗ ∗ − Re − 2 k V + − 2 2 M + , m X 4 3 T T spins M M ∗ V V,i ) ) 2 k 3 1 2 2 f 2 k m k 2 f f ∗ ∗ g ( ,µ m ( ∗ ∗ A – 38 – 8 m = m V T m ∗ M M  j ij , m ( V V πm M M + f f 2 , 2 1 3 + 2 F | 2 M ∗ f f ij − 4 2 6 1 ) 16 ∗ m N √ G ∗ k 2 M +( +( + + m , and ( m l 2 M } m λ 2 M m + + 2 = k + + m , m ∗ p ( − ,µ 4 1 2 ∗ m ( 1 | A 3 ∗ N k 4 4 M { ) j m ∗ M − k 2 M m m m  l = M Γ = ≡  m 2 i → 24 24 ) + 1 m , 2 3 √ 12 16 2 } N N ( − , m ≡ ≡ ≡ − ≡ − ≡ 2 2 → = 1 2 1 2 |M ∗ TT , m f ∗ ∗ ∗ ∗ ∗ − A VV, VV, VT, VT, 2 1 Z,µ ] and write the neutral weak axial current as are the masses of the final-state particles. The phase space function is f f f f J X spins m M 3 63 ( ( 1 2 λ m VV,TT,VT { × |M 2 = is the appropriate spin-averaging factor, and X spins n 2 ab 1 3 m The expression for the decay rates is given by Similarly, for processes involving vector mesons we find A.2 Sterile neutrino decaysWe into follow neutral ref. pseudoscalar [ mesons and defined as where where For sterile neutrino decay where JHEP03(2021)148 , η η f (A.17) (A.18) (A.19) (A.20) (A.13) (A.14) (A.15) (A.16) . For  π f , =  is given by s 0 3 µ π , f f . µ sγ q 0 = , θ + ¯ a M 0 2 6 d π if µ ! f cos √ 2 0 ¯ , 2 N dγ f 0 P . ) i ≡ ) m s ) c m − , q 5 η  s M ( 5 γ f w 8 − γ µ ! θ θ 2 0 µ = 0 and 0 1 a,P 1 2 , , 3 0 θ sγ √ c 0 θ

) ) 2¯ sγ sin η M √ π d d | sin 2 8 , + f | sin µ µ − + ¯ f 2 3 ) 4 cos 0 A a,µ . We then obtain e 0 ¯ ¯ d d d = M dγ dγ 0 j f 5 5 5 | f U = + f 9 0 | γ γ γ 0 − 8 6 0 − + π h 1 2 π µ µ µ 1 η 3 N f 8 8 √ u u . θ − θ ¯ ¯ ¯ 32 m µ µ dγ dγ dγ = 0 2 P  − 0 f − + + uγ uγ sin cos 0 , M π (¯ (¯ s , c , 8 2 F M u u u 8 – 39 – 8 f µ , f µ µ 2 2 f ) + 5 5 5 f f G q 1 1 c 0 γ γ γ 3 √ √ can now be written as sγ cγ 2 µ

1 θ µ µ µ M × √ 6 0 c . √ f P cγ = ¯ = ¯ = = = 0 (neglecting isospin breaking), but we do need to = 5 uγ uγ uγ i sin √ + (¯ (¯ (¯ γ ) M + ¯ 0 ,µ ,µ 0 c ! ( µ 6 3 2 V 0 J V V φ,µ 0 ω,µ η f 3 η 1 1 1 are given in table V M 0 ρ u 0 j + η j ) = 2 η j √ √ √ f j f cγ µ f f 0 e 8 P subscript labels the final-state pseudoscalar meson. Clearly , + i ≡ − ν 0 0 ) = = = = ¯ = = 8 8 uγ η 8 η M f is the mass of q (¯ f e θ ) f M ( 0 → ,µ ,µ ,µ ,µ ν M ( 0 P 3 A A A c  0 3 8 3

η f A η j j j 0 w a,P f N cos j √ m and → θ 8 mixing. We use the phenomenological parametrization in terms of 8 M 2 , f | 0 N 0 η θ - sin and the = A Z,µ Γ( , and η j 3 4 η } |  c f 0 π h − f , η 2 1 8 ,  3 2 or , is the Weinberg angle. We define the currents 0 = √ { w mesons, we have / θ c 0 V Z,µ = η η j The decay width of f a , 0 η We write the vector component of the neutral weak current as where where we add a 2f to account for the Majorana nature ofA.3 sterile neutrinos, Sterile neutrino decays into neutral vector mesons two mixing angles where the values of we have To clarify the notation:and for neutral pions take into account We define for where JHEP03(2021)148 . ,µ ) Ψ / (B.3) (B.1) (B.2) 2 N V J (A.21) (A.22) (A.23) j , m ) ) w − ) θ N 2 2 p 2 N ) l m − via a factor of p sin , l a − 4 3 2 p −  q 2 N − − 2 p 0 1 2 (( 2 N a ( p δ δ ) m m ) +( 2 l − 2 l . p − m V m φ,µ ( j 1 0 4 a,V − Ψ, respectively. We rewrite − δ  / 2 l 4 M 2 l w ) p   p θ ( π ( 2 δ 2 δ , 2 N a ) (2 ) 2 µ 2 2 m m sin  ]. We convert to four-dimensional , and J m a m 2 3 φ if , − |M| − 115 + ω 2 1 + 2 2 , = 0 0 is the mass of the decaying pseudoscalar , 1 2 ) p  i p X 0 ( ) ( and introduce the variable N − ρ δ 3 is the mass of δ 3 N M p ) l )  2 a p q,  π N m + N q − ( 2 p m p | l − (2 2 4 – 40 – 3 4 − p V ω,µ a d q 0 e 0 N a,V d j a p U − ( w | = M 2 Z 0 δ | πm θ 2 and a l p l g 2 N p Z p 32 2 V a,µ a 4 p 4 − 10 j 3 f d | sin d ) p 2 F 0 l ( π 2 h Z p 4 3 G Z 0 3 δ (2 √ 0 p 4 d 0 l p × 4 ) − p 4 . d π 2 l d ,µ p 0 (2 V ρ Z − Z Z 2 j q , and a function of four-momentum invariant scalar products. The denote the momentum of the decaying meson, outgoing meson, SM | 0 5 ) = 2 ) 3 = ) p ) Ψ. The decay width becomes w da N N M 1 π 0 / π θ 0 | p a,V p − p J

2 (2 Z (2 3 2 p M denotes the spin-averaged product of the leptonic and hadronic matrix 0 ) to obtain are listed in table d 5 0 e X integral by setting 2 ) 1 M p ν 2 ≡ a q 1 π 2 sin 2 2 g × , and |M| N q l , ω, φ, − p − → (2 p 0 Z |M| a ρ , (1 0 X ( N and 1 M Γ = 2 p 1 M P = 2 1 × a , 2 Γ( √ f a p da δ , where = 2 R q Γ = Γ = V Z,µ 1 = where for notationalperform convenience the we have omitted three Heaviside step functions. We on form factors are definedstandard below and techniques the and spin-averaged checked matrixintegrals with elements and are write PackageX calculated [ with where lepton, and sterile neutrino,meson. respectively, and element squared. It can be decomposed into a hadronic form factor, which only depends This work involves astraightforward but large tedious amount to evaluate. ofthese We three-body computations briefly discuss using phase-space here Mathematica.meson, computations, our the approach which decay In to are rate automize the is rest-frame given by of the decaying pseudoscalar where B Sterile neutrino productionB.1 in three-body decays Automizing three-body phase space integral calculations where j The hadronic matrix elements are defined as which correspond to the neutral vector mesons JHEP03(2021)148 ) B.5 (B.7) (B.8) (B.9) (B.4) (B.5) (B.6) (B.10) ,  2 ) and ( m , and, together . − 0 , a B.4  I a ) 2 N 2  +  m 2 2 ) , ,M − q 2 n M 2 · 2 l ) . l p m M p 1 2 · n ( a, m I l , ( + ) 2. The remaining integrals p = 2 − a ( n a / ( ) q 1 2 l  ≤ p n · λ only depends on 2 N c 1 2 · N 2 l 2 π am | m  p − =0 N X n 2 , p = )( p − M | · a   2 P M ( 2 l ) 2 l n ) p P 0 p m c − p da I and particle masses, and they also contain  = − − 2 2 =0 0 N − a ) 2 ) X n q a p ) q p N ( · should be evaluated via eqs. ( · m (  = − l m l − 2 δ l , + 2 p − ) – 41 – l p 1 M 2 l ( I a − m 4( M (  q m Z = δ − ) , p 1 2 − N 5 , 2 l 2 p )  2 l integrals. The last delta function gives the additional

) 1 = π m p 2 l 2 N | 2 N ( am q p (2  δ − m · 0 l M 2 I 0 2 , m | p 0 ) 1 M ) 2 l − 4 p q 2 q and 2 l ( d only. The final decay rate requires one remaining integral over a · a · δ X 0 , p p m a 0 p p Z a, m (  p ( )( a 2 4 + 2 Γ =  = q d / a m 1 · a n 1 l λ 3 I Z + p 1 2 2 π ( − a = denote the masses of the outgoing meson, lepton, and sterile neutrino, = . A straightforward calculation gives = = = − P } q N I 2 2 0 1 2 · I I I , l , m 1 M p ) are process-dependent functions of l , a 0 ( 1 2 { n c = m, m = 2 q p n · = 0 p We have automized thecases above when procedure no in or a justanalytically. simple few When hadronic this lines form is of factors possiblecases, appear, Mathematica we however, have the code. the checked integrals our integrals can results In are be with certain computed performed the numerically. literature. In most where the scalar productsand appearing are in thus functions of a for can be explicitly computed. We need and the form where the hadronic form factors. For our calculations we have relations These relations imply that the spin-averaged matrix element squared can be written in respectively. The hadronic form factor containedwith in three of the six scalar products can be taken out of the Here JHEP03(2021)148 , ∗  · , (C.1) p , with µ (B.14) (B.16) (B.11) (B.12) (B.13) (B.15) 0 β q P p 2 α M p m 2 . − q , . The pseudo- 2 µναβ µ  0 ) M 224 0422 p . . . 2 ) q 2  + p, ( 2 q 0 · ( µ q = 0 = 0 g 0 ) ∗ p | | 2  f + cb q µ us = ( q V + β V | ) , | µ − q  2 a ∗ α  P µ q  ) ( q ] in cases where a comparison 2 , 2 − . For a final-state pseudoscalar q ) ) + ( a 2 2 m m 0 116 , µναβ q f  + 2 ( m 2 − ) q T p 2 N, − m 2 m − f · q ] ) ( 2 + ∗ , µ − 2 − M 0  − q p  1 g µ M ( , ν e − , M + P m )( p + β µ ) f , respectively. A similar trick for the tensor 2 ) + ) q 974 00394  2 2 q 2 β 0 − . . q α q ( q p 2 ] P ( ν ( ) 0 P + ∗ α 0 0 + P/V ∗ ν p + a – 42 –  = 0 = 0 f m  a µ 117 | | and p 2 M p ( [ q + + 1 ub ud ) +  , q ) = µναβ p , 2 V µ → V ) m µναβ )  | 2 | · ∗ q M 2  2 ) i q  ( ( ) P q ∗ q 2 + ( ) 2 2 ( f  ( q 2 S  q M ( + S q f ( M ( f f 2 PS + ) = ig f g f 2 m = = = q = = = = ( i i i 1 + i i i i T ) ) ) ) ) ) ) 1 f p p p p p p p ( ( ( , , ( ( ( ( m P P P P P P P can be produced via the decay of a pseudoscalar meson, = M M M 218 997 M M M M | | | . . . Applying the equations of motion, the scalar form factor becomes | | | | 2 2 2 N µ 2 2 2 2 denote the mass of 0 q q q q q q q PS 1 p µ 2 5 5 = 0 = 0 f µ ¯ q µν γ | µν γ γ | | γ − 1 ) m σ 1 1 µ σ 0 ¯ q 1 cs cd ¯ q ¯ q 1 µ is a pseudoscalar or vector meson with mass | p γ ¯ | q | V ¯ V ) q ( | p 1 ) | ) 0 | | ) ¯ q 0 ) P 0 p is the polarization vector of the vector meson, and and | ,  ,  ( = p ) 0 0 0 P/V ,  1 ( µ M 0 p P 0 p µ ∗ h ,  0 ( ( P p 0  q m M ( 0 M 0 V p M V h M ( 0 V When a vector meson is produced additional form factors are required h M 0 M V h h M h M h We list all parametersmatrix we elements use are in extracted this from paper ref. [ in this section. The values of relevant CKM C Physical parameters, decay constants, and form factors where scalar form factor is given by is possible. where form factor gives which agrees fairly well with lattice computations of ref. [ where A sterile neutrino mass where meson, we require the following form factors: B.2 Definition of three-body decay form factors JHEP03(2021)148 (C.3) (C.2) , . Parameters , 8  K ) 2 MS q ] ] ] ] . = 93 MeV ( c z η s  119 119 120 122  k w K ] to parameterize the form [MeV] w θ and θ K 2 w 0 V 2 − M θ 266 [ 308 [ 230 [ 209 [ η w k f 126 sin 2 θ , , sin 1) η 2 2 3 a = 2 GeV in 4 3 63 − g V ( + µ sin 2 sin − ] ] ] 1 2 M 2 3 − . 1 2 − , m ] ∗  ∗ ∗ ] s k − 123 123 124 ρ 2 )) D D , K of neutral vector mesons. 123 2 2 123 [ k √ [ q a ) √ ( ◦ g meson 2 ◦ z 7 MeV 18 GeV q . . ( ( ]  ] ] z 2 -6.9 ] ] ] ] -21.2 0 k + k 148 GeV [ 165 GeV [ 335 GeV [ – 43 – . . . = 4 = 4 b b 0 0 0 118 118 1 1 b d 118 118 118 121 − − =0 =0 [GeV 1.29 [MeV] k k X X 0.171 1 0.155 - 0.232 c K K 0 8 0 8 0 η a P θ θ f f f f M 212 [ 249 [ 187 [ 434 [ M f 155.6 [ 130.2 [ 2 pole 2 pole → ] ] ] M /m /m ] 1 1 ) P 2 2 s ( 64 q q 64 64 . Decay constants and angles for [ M [ [ 125 , m B    . Decay constants and   s , m 0 − − c  φ ω ρ J[ π 1 1 B B D D K meson meson Table 9 ) = ) = . Decay constants for charged pseudoscalar and vector mesons. 2 MeV 27 GeV 2 2 . . q q Table 10 ( ( 0 , respectively. + = 2 = 1 f f c u 10 Table 8 m m and 9 Decay constants for pseudoscalar and vector mesons are given in tables We use the quark masses at a renormalization scale of tables C.1 Form factors for We apply the Bourrely-Caprini-Lellouch (BCL)factors, method [ to calculate decay constants for the neutral pseudoscalar and vector mesons are given in JHEP03(2021)148 (0). 0 (C.4) (C.5) (C.6) f (0) = + f transitions from transitions from π π → → B 86 ]. We write 1 , . 2 . D b = 3 and 66 0 1 i − 127 and K (0) z and ] D 2 2 )+ , 2 11 45 68 61 K 828 . . . . 1 → q . 197 ( b 0 0 7 2 0 1 . 0 0 0 V t t z . 224 1314 0342 0 . B − − − − → . . [GeV − 0 M 0 0 − − . We set P (0) D 2 1 + + + ). To better present these form factors z ) t t ) h 0 1 2 0 . 49  b m √ √ 909 794 360 233 404 . b . . . . . 0 2 √ . 0 0 0 0 0 P q − B.16 − + 066 084 985 188 11 (0) c . . . . 0 0 z 2 2 0 2 1 1 − − b (0) 12 q q − − − − z − 65 − (1 . M − − ) – 44 – 2 325 325 √ + + . . q t t (0) = 5 ( )( [GeV] decaying into z ) + (0) p p 7647 7647 6117 6117 ∞ ∞ ∞ = 5 = 5  m + f . . . . f c 0 0 0 0 ∗ ∗ D (0 pole + ∗ B B = 0 ) through eq. ( P ) = B m 0 2 m m 2 2 π π M b K K m q M q and ( ( → → (0) + → → f z  ) f D D + 0 D D → = ( + 0 s ) ) a f f ( s s f f ( ( 0 = K K t B D π π D D 0 transitions, we use the methods of ref. [ → → → → → → / f s s ) ) B B + 0 s s + ) and B B ( ( + 0 f f K and 2 f f f B B + 0 q 2 f f ( through ) → f 0 2 m b D + and M ]. ) is the function 2 π ) contains seven form factors which must be determined. The pseudoscalar form . Best fit parameters for the form factors in . Best fit parameters values for the form factors in q = ( ( 127 ]. , → z + t 63 B.15 128 D Table 11 ref. [ and list the best-fit parameter values in table C.3 Form factors for Eq. ( factor is related to For Table 12 refs. [ where This determines The best-fit parameter values are given in table C.2 Form factors for with JHEP03(2021)148 ] V 129 (C.7) (C.8) (C.9) ) for ]. . ]. − ) 2 , (1 q 129 ) 129 ) , ( V , 3 2 0 T q 63 g 23 10 46 42 42 63 M . . . . . ( (0) ( 23 19 33 31 45 q 1 3 1 1 1 1 1 . . . . . + · T 0 0 0 0 0 σ 2 a f can be produced. ) , P and , ) ) m φ 0 ) 2 − 2 A T 02 38 64 74 98 + ) (0) q . . . . . 4 ( 32 27 68 66 78 pole 2 2 . . . . . ( 1 0 0 0 0 0 T q 0 0 0 0 0 and σ ) from refs. [ + M ( f ) from refs. [ ( g /M ∗ − ) for 0 4 − − g . C.9 ) q − C.9 K 1 2 4 V = (0 T σ , 25 44 57 60 66 (0) )–( 32 27 68 66 78 . . . . . ( )–( 1 0 , we use the simpler form [ . . . . . ) = ) = P 1 1 0 0 0 0 0 T 3 0 0 0 0 0 2 2 + σ /M f K q q T C.8 M 4 C.8 ( ( q , 2 3 0 is only induced by SM weak interac- 2 2 pole ) η 2 σ ∗ s and (0) , A 67 89 40 40 04 32 26 75 24 62 49 49 2 ...... /M ...... + ( 2 D η 0 0 1 1 1 1 2 0 0 0 0 0 0 1 ,A A T ,T 2 (0) (0) q f V σ , . 1 →  f f ) m 2 σ q 2 s /M · A ) q + 15 2 B 1 ( − , (0) q P 29 70 26 66 59 66 – 45 – f 1 A 30 50 78 73 86 meson, 1 ...... 1 463 ) . . . . . ( M . 0 0 0 0 0 0 A s A 2 σ 0 0 0 0 0 1 )(1 s 0 f q σ D ( − D ], we define the following dimensionless combinations 2 ) = + pole 1 2 mix with each other and we can consider them as the a ) to table q 0 0 129 (0) ,T ( 37 67 30 69 66 76 /M + 0 ...... η A 17 36 59 54 60 1 ) 350 13 . . . . . 2 ( ) = 0 0 0 0 0 0 A . 2 2 0 0 0 0 0 q 1 2 f 0 q q σ q is the pole mass, which is ) ( ( − 2 and − f q g ( M ) η (1 9 q (0) 38 95 31 76 03 ,A . − ...... V 2 · 27 46 57 59 66 0 ) V ( 0 a 372 0 0 0 0 1 . . . . . q . 2 f 1 0 0 0 0 0 A 0 P ) = q σ ( 2 ) + q g − ∗ ∗ s 2 ∗ ( ) ∗ ∗ ) ∗ s q ρ ρ ∗ and ∗ f D K D ( 2 K m ρ ρ D K q 1 D K f → (0) → ( 1 →  → T + → → . Part I: Best-fit parameters values for eqs. ( → → f . Part II: Best-fit parameters values for eqs. ( σ → + → s , s → → B D g B s m s D 1 B B D B M V 2 B D − B B = ) = ) = ( 0 . For the remaining form factors 2 2 1 Table 13 Table 14 A q q T ( ( 2 V T fit parameters are given in table C.4 The semi-leptonicThrough decay of the semi-leptonicThe decay pseudoscalar of mesons In all scenarios we consider,tions the and transition we do not list form factors associated to BSM currents. The values of the best to describe and and choose the following three-parameter formula and follow the conventions of ref. [ JHEP03(2021)148 , T D f m 01 . For 112 415 325 331 331 ...... ∗ 3 (C.10) (C.11) 46 34 45 = and . . . T D 0 1 0 = 2 P = 2 = 5 = 5 = 6 = 6 (GeV) + ]. ∗ V f ∗ ∗ ∗ ∗ ∗ s s c c /m M D ∗ s B B B B D 2 3 M 77 02 01 45 08 68 ...... m D T T m m m m m 129 ] 0 0 2 0 1 0 , ]. ]. m 63 with , 129 = , , 129 1 87 968 ) 129 367 275 275 77 25 279 . 0 . 06 . . . . . ) s e . T 2 . 71 44 V e 0 0 . . η ν 0 T C.3 ν ) 0 0 . All the related best φ = 1 = 1 M (GeV) − = 5 = 6 = 6 = 5 + ψ φ + P s c c s 2 D e 47 63 → B 0 B B . . B D e A M m 0 0 → m 1 m m m m 71 22 0 with . . T s Ds ) + ) from refs. [ 0 0 0 ∗ ) + ) + cos( 0 η D 1 η 3 f 64 29 0 K n . . T C.9 A 62 51 50 17 34 m m 0 0 η . . . . . ( ( 2 ( ) 2 0 0 0 0 0 42 58 for → s 0 when calculating the decay width. s decays from ref. [ . . )–( σ decays from ref. [ A ψ η η 0 0 ) ∗ s s 0 0 s ( 73 10 ) 0 D Ds . . η C.8 D A 2 → D → 0 0 T m 90 19 50 80 0 1 m s . . . . ( s 57 29 42 . . . 2 0 0 0 1 = sin( A ]. The decay rates are [ D = 0 0 0 D σ as 0 10 26 0 . . V s V ) 1 0 )Γ( η 131 )Γ( 1 0 – M T ψ ) to parameterize the form factors 67 20 31 ψ 0 0 0 0 0 . . . ( ( A ( , 2 0 0 0 2 2 s – 46 – ) to parameterize T 94 24 ) , η σ 0 129 . . 0 C.8 f D [ ] s η 0 0 ) η ◦ . m 2 C.9 ) m 0 ( V 129 A 54 50 41 16 ψ s [ 0 1.04 0.24 070 . . . . = 0 ( 78 21 76 . 17 ) = sin ) = cos η . . . 0 0 0 0 2 f 0 e e 0 0 0 P σ ¯ ss ν ν sin( → ), which permits any use, distribution and reproduction in ) M s T 77 24 ≡ + + and 0 1 − . . f + D s e e 0 0 0 A 60 10 20 n f 0 0 510 . . . η ( 0.78 0.23 16 . η 0 0 0 2 , we use eq. ( K + + 0 ) is around 40 0 σ 0 , and η 7 ψ 0 η 72 41 . → ∗ 0 . . ) ψ f T and , and use eq. ( 80 24 0 ) 0 . . 0 0 s /K 0 f ∗ η → K 0 0 s → . Part I: Best-fit parameters for A 16 . Part II: Best-fit parameters for D ¯ D 0 0 0 0 dd η s 600 CC-BY 4.0 . m ( 2 s . ( 0 2 + → 0 D √ s = cos( σ D + 72 20 0 /m 0 → η 78 33 38 This article is distributed under the terms of the Creative Commons . . s ¯ uu f . . . f ∗ s η 0 0 Γ( . Part III: Best-fit parameters for eqs. ( ) s 0 0 0 Γ( D D → ≡ V D 30 ) means we consider the mass of 0 0 0 0 ( s 561 m . ) . n 2 Table 16 0 0 Table 17 1 2 for ( + 0 D 78 23 η 0 σ (0) . . = η σ σ f 0 0 ∗ f ∗ ∗ s m ∗ ∗ D ( ρ ρ Table 15 D K s pole D K pole m 1 2 η (0) → → σ σ M → → → → = f , m s s 2 B D B D σ V B B M fit values are given in tables Open Access. Attribution License ( any medium, provided the original author(s) and source are credited. where For the decay with the remaining decay channels, we can use the same method as that in where the mixing angle mixtures of JHEP03(2021)148 22 (1977) decays (2009) 0 Conf. 22 (2017) (1975) 7902131 New J. , 67 670 Z , 481 (1974) 275 01 11 ]. 10 Phys. Rev. D , JCAP Prog. Theor. Phys. ]. , , Sterile neutrino Dark SPIRE ]. Phys. Rept. IN Phys. Lett. B Phys. Lett. B Conf. Proc. C , , , , ][ (2008) 105 Phys. Rev. D SPIRE ]. (2016) 1 , IN Physics of leptoquarks in SPIRE Phys. Rev. D Int. J. Mod. Phys. E ][ 466 , IN , 641 [ ]. SPIRE x U(1) Theories ]. 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