Jhep03(2021)112
JHEP03(2021)112 Springer March 10, 2021 : January 18, 2021 January 31, 2021 : : Published a,c Received Accepted , Published for SISSA by and Zhen Liu https://doi.org/10.1007/JHEP03(2021)112 b [email protected] , Roni Harnik b [email protected] , . 3 conversion. If the compositeness scale lies at or below a hundred Patrick J. Fox, e a 2012.01443 → The Authors. We consider a class of models in which the neutrinos acquire Majorana µ c Beyond Standard Model, Neutrino Physics, Technicolor and Composite Mod-
, [email protected] and eγ → E-mail: [email protected] Maryland Center for FundamentalUniversity Physics, of Department of Maryland, Physics, College Park,Theoretical Physics MD Department, 20742-4111, Fermilab, U.S.A. P.O. Box 500, Batavia, ILSchool of 60510, Physics U.S.A. andMinneapolis, Astronomy, MN University 55455, of U.S.A. Minnesota, b c a Open Access Article funded by SCOAP Keywords: els ArXiv ePrint: µ MeV, the rate for neutrinolessreduced double beta by decay an is order suppressedgive of by rise form magnitude to factors or spectral and more. distortions mayto be in The be the late observed cosmic decays in microwave background of future that experiments. relic are singlet large neutrinos enough can consequence of the scaling dimensionsthat of this operators class in the ofincluding conformal models colliders field and theory. has beam We interestingdouble show dumps, implications beta searches for for decay, a leptonscenario and wide can flavor cosmological give violation variety rise and observations. to of striking neutrinoless of experiments, signals involving hidden At multiple sector displaced colliders vertices. states and The can exchange beam lead dumps, to observable this rates for flavor violating processes such as coupled hidden sector.particles that In obtain this theirthe framework, masses scenario the in through which light theand the neutrinos inverse the strong seesaw are dynamics compositeness mechanism. is partiallythe scale approximately Lagrangian composite We lies conformal necessary focus at in to on realize the or the ultraviolet, below observed the neutrino masses weak can scale. naturally arise The as a small parameters in Abstract: masses through mixing with singlet neutrinos that emerge as composite states of a strongly Neutrino masses from low scale partial compositeness Zackaria Chacko, JHEP03(2021)112 ]. 16 – 16 14 23 5 33 32 ]. Since the compositeness scale arises 19 – 17 – 1 – 27 ], and the Majoron model [ 13 38 – 1 23 25 30 12 19 decays 21 34 1 37 Z 30 11 and W One attractive class of models that can naturally explain the smallness of the neu- 7.3 Spectral distortions of the CMB from late decays 7.1 Cosmological history 7.2 Cosmological bounds on the compositeness scale 5.1 Muon magnetic5.2 moment Lepton flavor violation 3.1 3.2 Current constraints3.3 and future reach Novel from signatures HNL searches its various different incarnations [ trino masses are those incomposite which states the of neutrinos a acquire stronglyfrom mass dimensional coupled transmutation, through sector it their [ can couplings be to parametrically lower the than the Planck scale. The 1 Introduction Over the last few decadesnos a series have of tiny experiments but have non-vanishingever, conclusively masses the established that and mechanism neutri- have that determined givesmany their rise well-motivated mass proposals to splittings. these that small have How- been masses remains put a forward mystery. are Among the the seesaw mechanism in A Matrix elements and widths 8 Astrophysics 9 Conclusions 6 Neutrinoless double beta decay 7 Cosmology 4 Beam dumps 5 Muon magnetic moment and lepton flavor violation 3 Colliders Contents 1 Introduction 2 An inverse seesaw model from strong dynamics JHEP03(2021)112 . 1 ] and . 22 – 20 ]. The difference be- 26 GNGOGPVCT[ǁ , 25 ]. The cosmological signals of this 23 ]. – 2 – 24 XKQNCVKQP .PWODGT EQORQUKVG0 . . Lepton number is also broken by dynamics in the hidden sector. Λ OKZKPI ]. The composite singlet neutrinos are analogous to the vector-like fermion , lies at or below the weak scale. This allows the possibility of directly producing ]), in which the SM fermions receive their masses through partial composite- Λ . A sketch of the generation of neutrino masses in our framework through partial compos- 30 , GNGOGPVCT[EQORQUKVG 28 , 29 GNGOGPVCT[ǁ 27 The spectrum of the neutrino sector consists of the light neutrinos, which are now In this paper we focus on the scenario in which the compositeness scale, which we In this paper we explore a class of models that realize this scenario and show that they of a low scaledimensions seesaw of model operators can in be the understood CFT. here aspartially naturally arising composite from and the parametricallyfinite scaling lighter tower of than composite the neutrinos compositeness of scale, which and the an lightest in- states have a mass of order the denote by composite states at colliders andcoupled other hidden experiments. sector For to concretenessbroken we be take at a the the conformal strongly scale fieldThe theory small (CFT). parameters The required conformal to symmetry obtain is the tiny neutrino masses within the framework see [ ness [ resonances in composite Higgs models.els, However the here, strongly in contrast coupled toto sector composite consider is Higgs compositeness mod- neutral scales under that the lie SM well below gauge the groups electroweak and scale. so we are free From the perspective ofneutrino the masses low arise energy through theory,tween the our at framework inverse or and seesaw that below mechanism ofare the [ a now conventional compositeness inverse the seesaw scale, hadrons isite the that of the a particles. singlet new neutrinos strong Our force, approach and is the similar light neutrinos to are that partially of compos- composite Higgs models, (for reviews class of theories have been considered in [ can lead to richneutrinos phenomenological acquire consequences. Majorana masses Wecomposite through consider states mixing a with of framework singlet a in neutrinos strongly which that coupled the emerge hidden as sector, as shown schematically in figure smallness of neutrino massession is greater explained than by the four.particles. fact In they The this arise composite scenario from neutrinoto operators the framework the origin has of neutrinos of been dimen- could the linked be baryon to asymmetry either of dark Dirac the matter universe or [ [ Majorana Figure 1 iteness. Elementary neutrinos mixin with the composite composite singlet sector. neutrinosthe The and light elementary lepton neutrino number and mass is maysector, composite violated be with neutrinos, explained a either or continuum by of by small possibilities mixing small in between lepton between. number violation in the composite JHEP03(2021)112 boson ] that 44 W ]). that blend into a 46 Λ E ] to the two-brane Randall- ] (see also [ ]. It was shown in [ 45 38 , 43 – 37 40 . We now consider each of these different 3 ]. The spectrum of particle masses in this the spectrum of composite states blends into WPRCTVKENG7 EQORQUKVG00 RCTVKCNN[EQORQUKVGǁ 32 Λ , ] relates strongly coupled CFTs to theories of – 3 – 31 36 – 33 ], originally proposed as a solution to the hierarchy problem %(6 39 . 2
. At scales well above 'PGTI[ Λ this scenario can give rise to striking signals at colliders. A ǁ Ʃ O . A sketch of the spectrum of particle masses in our framework, showing the light partially LHC signals: produced at the Large Hadronone Collider or (LHC) more can now composite decay singlet into a neutrinos. hard lepton The and singlet neutrinos subsequently decay The class of models that we are exploring has interesting implications for a wide variety The AdS/CFT correspondence [ • different types of signals isexperimental summarized in probes figure in turn, along with the constraints from current data. more closely related to our construction was considered inof [ current andfor near-future lepton experiments, flavor includingobservations. violation colliders and and The neutrinoless beam range double dumps, of beta compositeness searches decay, scales as that well can as be cosmological probed by each of these with composite states playingshare the many features role of of ourtheories the construction. the singlet However, SM an neutrinos. leptons important arescale difference themselves These is is partially theories that then or in therefore constrained entirely these composite.different. to lie The A above compositeness a holographic TeV, model and consequently of the neutrino phenomenology masses is from very low scale compositeness that is Sundrum (RS) construction [ of the SM.RS Several solution authors to have exploredthe the neutrino AdS/CFT hierarchy masses correspondence problem, within relatesto for the theories a example framework in certain [ of which class the the neutrino of masses these are extra generated dimensional by the models inverse seesaw mechanism, compositeness scale a continuum spectrum ofscenario “unparticles” is [ shown in figure gravity in higher dimensions.conformal dynamics Theories is of spontaneously phenomenological broken interest are in dual which [ the strong Figure 2 composite neutrinos and thecontinuum composite of resonances “unparticles” at at the higher compositeness scales. scale JHEP03(2021)112 [GeV] Compositness scale ⌫ 2 . 10 5 conversion, will be able to probe this decays µ e 10 2) → µ g ( W, Z, h – 4 – 1 in general the couplings of the SM to the hidden decay if the singlet neutrino mass lies below a few GeV, this experiments searching for neutrinoless double beta de- e D 1 . ! 0 . µ 4 decay: . β 2 decay 3 K 10 decay 3 conversion -less 10 e ⌫ SN1987A ) ↵ e decay ! N . A sketch of the various probes of neutrino compositeness and the range of compositeness µ ⇡ ( Neutrinoless double cay are extremely important becausethat the neutrinos observation are of this Majoranaated process with particles. would neutrinoless double establish The betawhich characteristic decay is is momentum set of scale by order the associ- nucleon the spacing pion in mass. the nucleus, It follows that if the scale of compositeness in the sector violate the flavorthis symmetries scenario can of give the rise toof lepton lepton hidden flavor sector violation sector of at loop states. thefor level SM. through the The the lepton exchange COMET Consequently, flavor andclass violating Mu2e of process models. experiments, We which discuss are this searching in detail in section signals, particularly in thefind case that when DUNE these andfurther decays SHiP detail result are in in both section displaced sensitive vertices. to this We Charged scenario. lepton flavor This violation: is discussed in discussed in section Meson decays in beam dumps: class of models candumps also can now be decay into discovered finalsubsequent at states that decays beam contain of one dumps. these or more singlet Mesons singlet neutrinos neutrinos. produced into The SM at final beam states can lead to striking timescales, giving rise to eventsthis with striking multiple signal displaced would vertices.conventional allow inverse The seesaw this observation models class of in of whichfind the theories that singlet the to neutrinos HL-LHC are be will elementary. distinguished haveFASER-2, We excellent and from sensitivity MATHUSLA to more can this class all of further models. improve Codex-B, on the HL-LHC reach. This is into SM final states. In much of the parameter space, these decays are slow on collider Cosmology • • • CMB Spectral dist. Figure 3 scales that they are potentially sensitive to. JHEP03(2021)112 (2.1) below Λ . 6 . The corresponding S O . 9 . S O S λ + . 7 CFT – 5 – ⊃ L UV L that will play the role of singlet neutrinos. We will return c N gets large, it triggers breaking of the CFT at a scale which we . S 8 O if the compositeness scale is lower than or of order 40 MeV, the temper- corresponds to the mass scale of the lightest composite particles. We in the early universe, thermal contact with the SM ensures that the hid- Λ . We focus on the scenario in which there are no pions or other light composite Λ discussed in section Supernovae: ature in the coreOnce of produced, a they supernova, get hidden trappedcontribute sector to inside energy states the loss. core can and The bebe thermalize, presence produced expected and of at so these to the they additional affecttainties degrees core. do in the of not supernova freedom supernova dynamics would dynamics. we cannot However, robustly given rule the out present this uncer- possibility. This is If the compositeness scalecan give lies rise below to 50 spectral MeV,are distortions these in potentially late the observable cosmic decays microwave inclass background into future (CMB) of charged that models experiments. states is discussed The in cosmological section history of this abundance is exponentially suppressed. Ifthe the temperature compositeness at scale which lies below theto an weak the MeV, interactions total decouple, energy this densityan in results MeV neutrinos in are that a therefore is correction disfavored. unacceptablycay large. The into remaining final Values relic of states composite that, states in eventually addition de- to neutrinos, may also include charged particles. decay is suppressed byconsequences for form experiment. factors We and discuss this may in beCosmology: detail greatly in section reduced, withden sector important is populated. Atthe temperatures hidden below sector the annihilate compositeness efficiently scale into the light states neutrinos, in with the result that their neutrino sector lies at or below a hundred MeV, the rate for neutrinoless double beta In the next section we present the framework that underlies this class of models and and their Dirac partners • • denote by states, so that require that the spectrum ofN light composite states containsto at the least issue two of pairs flavor of at fermions the end of this section, but for now we suppress all flavor indices. of a conformal field theoryterms (CFT) in deformed the by Lagrangian a relevant take operator the form, When the deformation 2 An inverse seesaw modelIn this from section strong we describe dynamics ourcompositeness framework of for neutrinos. neutrino mass For generation concreteness, based we on will the partial take the strong dynamics to be that determine the allowed parameter space.various types In of the signals subsequent as sections listed we above. discuss each We conclude of in the section JHEP03(2021)112 ˆ λ ], the 47 (2.4) (2.5) (2.6) (2.7) (2.8) (2.2) (2.3) 2 . The / 2 5 / 3 ≥ ≥ N N ∆ N ∆ − ∆ 2 / . 5 µ ) . p ) . i µ c . 1) . σ − ) − 2 π p + h N N − ( N c ) .. ) c π , N . 2 is a primary operator of the CFT N / N N 2 1 N / . π 2) Γ (2∆ − . 3 + h 2) cos(∆ c M O / . N , unitarity requires ( / − Λ Γ (∆ 3 N ∆ 3 N N − ) gives rise to a term in the low-energy O cos (∆ ∆ A − O + h i − c . We now assume that at high energies, in being a free fermion. For 2 N 2.3 N 1 N LH . Here N µ is given by, = − UV 2 Λ ∂ (∆ / O i λ 3 µ (1) Γ (∆ M 0 – 6 – | ¯ s − σ C λLHN 1 is of order O c ˆ λ N i − 2) N 2 ¯ ˆ λ ⊃ ∆ UV N N N / represents the ultraviolet cutoff of the theory, and / . We use the two-point function to normalize the i ∆ λ 5 (0) 3 2 M M − IR 2∆ π † N C / , using the methods of “naive dimensional analysis” 2 + ) UV − 5 L / N O ) respect a global lepton number symmetry under which π has charge 16 ⊃ 3 ∼ ) N N ) M O x (2 < µ λ π ( N 2.6 ∂ UV to be of order, N (4 N µ = L Γ(∆ ¯ σ corresponds to O λ 2 ∆ h / and ¯ = N 1 2 i T ≤ / − λ | N +1 0 ⊃ C 2 , the interaction in eq. ( h ∆ / Λ = 3 ) and eq. ( 3 A IR ) makes the theory ultraviolet sensitive, and additional counterterms ipx L N 2.2 , takes the form xe 2.3 ∆ 4 Λ d ] we estimate Z , following the conventions of unparticle physics for fermionic operators [ 50 N – carry charge O is its scaling dimension. Unitarity places restrictions on the scaling dimensions c 48 N N ∆ The CFT is assumed to couple to the SM through a neutrino portal interaction which, and The terms in eq. ( L (NDA) [ where the order one multiplicative factor At energies of order Lagrangian of the form, Based on our normalization of where coupling in eq. ( involving the SM fieldsscaling are dimensions, required foroperator consistency. We therefore focus on the range of In this expression the massis scale a dimensionless parameter takenand to be of CFT primaryLorentz operators group. which depend Forlimiting spin on case 1/2 their of operators transformation such properties as under the above the scale singlet neutrinos, We expect that the mass parameter The low energy Lagrangian then includes kinetic terms and mass terms for the composite JHEP03(2021)112 , c ), 1 2N ≥ (2.9) O 2.10 c (2.16) (2.12) (2.13) (2.14) (2.15) (2.10) (2.11) 2N ∆ . , c . . c 2N . ) ∆ 2 − p + h 2 ( ) θ 2 1 ) c i c 2N c − N ∆ is a dimensionless param- ( 2 2N − p c c ∆ 2 2 ], ˆ ) µ µ − A 2 ( . p 51 ) + , , ( . ) diverges in the ultraviolet, and 1 π c 4 2 1 2 . c ) to normalize the operator − , and − . c c c 2N c = c 1 2.10 . . Assuming this deformation carries + h 2N . 2N as per the conventions of unparticle N 2N 2.11 1 2N EW c ∆ ∆ λLHN c O M abs ∆ + h A
> λv 2N + 2N i 2 sin (∆ c s 0 O the Lagrangian contains a lepton number ) i | O UV c c 4 N Λ 2 c 2N i 1) c µ Λ − ], M N ) simplifies to [ c 2N − ), the Lagrangian contains a small deformation ∆ ( N − c (0) = ∆ – 7 – 2N c ˆ c c 32 µ N = 2 − , 2.3 Λ µ ∆ UV 2 2.10 i ) is only valid for the range of scaling dimensions c 2N 2N M ) ) that the low energy Lagrangian contains all the 0 31 ˆ | M µ ⊃ π O seesaw µ . i ) corresponds to a free scalar. For the range of scaling − (4 2.10 ⊃ x C IR Γ(∆ 2.12 is given by, c ( inv (0) | L c c ∼ µ N = ν = 1 UV † 2N c µ C 2N ) and eq. ( c L µ m ∂ µ O , which explicitly violates lepton number, O µ C h c 2N ) , the left-hand side of eq. ( ) and ( 2.1 ¯ σ 2 x c we get a contribution to the masses of the light neutrinos from T , we can normalize ∆ 2N | ( 2 ¯ c c 2.6 , at scales of order 0 N ≥ O is related to the parameters in the ultraviolet theory as h i N 2) † c 2N < c ), ( + ipx − µ O c 2N ( h xe N 2.2 ∆ 2N and 4 T µ | d ∆ ∂ 0 N µ h Z ¯ σ ≤ . For ipx ¯ 2 N 1 i is the scaling dimension of the operator xe < 4 c ⊃ ), is ultraviolet safe. We therefore use eq. ( d c 2N IR 2N Z ∆ L 2.11 ∆ ≤ Integrating out the inverse seesaw of order, We see from eqs.ingredients ( necessary to realize the inverse seesaw, where the multiplicative factor a lepton number of violating term of the form, The mass parameter The operator normalization in1 eq. ( we can no longereq. employ ( this normalization.over However, the the entire absorptive range part of of scaling eq. dimensions ( The absorptive (imaginary) part of eq. ( eter. The unitarity bound onwhere the the scaling limiting dimensions case ofdimensions scalar of operators restricts physics, this time for scalar operators [ of the CFT, denoted by Here addition to the terms in eq. ( JHEP03(2021)112 λ . We (2.18) (2.19) (2.17) 8 . There . ≤ λ 2 c # 2 2N / 3 may be playing − c N + ∆ ∆ N 2N O ), does not contribute 2∆ UV Λ , c M 2.12 2N ). O ) cannot be larger than the Λ EW 2.17 2.12 and v . N . Therefore, in this framework, the ), the operator . c ˆ N λ O c λ . 2N M C shown in eq. ( ν 2.18 EW O ) that the sizes of these effects depend on v c + h m ), we expect an ultraviolet contribution to #" . This term, although very similar in form 2 4 N c √ − 2.9 µ and 2.17 N c that diverges in the ultraviolet translates into & 2 µ – 8 – 2N N 2 λ ∆ ) O ⊃ & will also generate a lepton number violating mass ) and ( IR LH c ( UV N L Λ , and is given by eq. ( ) controls the strength of lepton number violation in the EW 2.3 2N Λ M M v O c . From dimensional considerations we see that the condition 2.17 from the condition that the Dirac mass for the neutrinos i ˆ . The size of this effect is controlled by the CFT three-point µ ) 2 µ λ z . This translates into a lower bound on the coupling that triggers the breaking of conformal symmetry. ) ( C c Λ S ), we have assumed that the contributions to the neutrino mass " , O LH 2N Λ . Λ ( ) be less than the compositeness scale. From this it follows that c O 2.17 . We expect comparable but slightly smaller contributions to the dominate over the contributions from higher scales. In general, in the ) ∼ µ y sits at its lower bound in eq. ( Λ 2 2.6 ( is expected to be of order λ N O µ N ) EW x 174 GeV M ( λv at the scale N ≈ c N hO µ EW v ∼ In general the deformation Another restriction on the parameter space arises from the condition that the Ma- In obtaining eq. ( ν m term for The parameter to the lepton number violatingsignificantly mass to term the for neutrino masses. In the limit when the role of the operator jorana mass for thecompositeness scale, singlet neutrinos arisingis from also eq. an ( upperobtained from bound eq. on ( must lie in the range that there not bea a bound contribution on to theassume scaling that dimensions of this the boundmasses operators arises is from satisfied, scales of so order that the dominant contribution to the neutrino from scales of order presence of the deformations eqs.the ( Weinberg operator function composite sector or byof the possibilities smallness in between. ofthe We elementary-composite see scaling mixing, from dimensions with eq. of ( ascaling the continuum dimensions operators of CFT operatorssmallness can of provide neutrino a masses. simple and natural explanation for the The first square bracket in eq.composite ( sector. The secondSM square neutrinos bracket controls mix the withpresent degree the to to which composite generate the singletexplained Majorana elementary in neutrinos. neutrino this masses. Both framework of either The by these lightness the effects of approximate must lepton the be number neutrino symmetry may in be the neutrino masses from integrating out theour higher final mass singlet expression fermion for resonances. the Therefore masses of the light neutrinos takes the form, with JHEP03(2021)112 = α (2.22) (2.23) (2.20) (2.21) with . We work c α 3 ) now takes , . We further N 2 N , # . ∆ 2.15 . In the limit of c . ) as, and = 1 U i α + h . These nonrenormal- N 2.22 2 c β ) N π c α (4 N in the CFT. Including this ..., α N αβ .. ) + 2 c O c . 2 µ ) ( 2 + h N . + c Λ α N α N O ( α EW 0 N v H κ i HN i M c iα L α + are the lightest states in the hidden sector, so L λ 2 N 2 / c α iα 3 µ – 9 – = λ diagonal. In this basis, the SM neutrinos will, in ∂ − N iα i N µ N 2 ` ˆ + µ λ N ¯ σ ∆ α UV Λ c β α N M and is also expected to contain four-fermion interactions ¯ ¯ M are expected to be of order Nσ N N U α i 0 c α Λ κ ⊃ N κ N + α UV αβ and N ) ⊃ − L . We therefore expect that there will be both flavor preserving µ κ N α ∂ IR µ and in general there may be more. Reinstating the flavor index M N L ( ¯ σ c " α ¯ N − i ⊃ N,N IR ’s. These take the schematic form, L operators so that they all have the same scaling dimension, N , with a corresponding number of operators may be rotated to make , so that all the composite singlet neutrinos are approximately degenerate. For N α S N c α N O O ,N α N Keeping track of flavor indices, the low energy Lagrangian in eq. ( To generate the mass splittings necessary for neutrino oscillations there must be at The Lagrangian at the scale ,..., 2 , different assume that this symmetryoperator is preserved inconcreteness, the we assume breaking that the ofthat conformal their invariance only by kinematically allowed the decay modes are to SM particles. In order to relate elements of the mixing matrix are related to the parameters in eq. ( In our study of theassumptions. phenomenology of this We class assume of that models, there we will is make a certain simplifying discrete symmetry in the CFT that relates the The general, couple to all the and flavor violatingneutrinos processes is involving conventionally the parametrizedsmall heavy in mixing neutrinos. terms between of the a The active mixing mixing and matrix between the the composite singlet neutrinos, the corresponding The flavor index on thein lepton the doublets runs basis over in the which SM the generations, charged lepton massesth are form, flavor diagonal. least two pairs of that we have so1 far suppressed, the singletflavor neutrinos index come the couplings in between pairs thegiven SM by, lepton doublets and the CFT operators are now symmetry. The parameters izable terms are characteristicnot of required the to composite realize the nature inverse seesaw,of of they this the play an class singlet important of neutrinos. role models, in the Although particularly phenomenology in the context of astrophysics and cosmology. between the where we have shown two such terms. These interactions respect the overall lepton number JHEP03(2021)112 , , . c c c ˆ Λ µ µ 2N and ∆ N 1 05 08 05 . [ eV] 045 . . . . O ν 0 vanishes, and m ˆ N λ is small and ∆ 2 ˆ λ − 7 0 5 0 80 0 300 0 [ MeV] or 10 c c , which control the ˆ µ c 8 7 8 7 8 2N − − − − − are of order one, with the O 10 10 10 10 10 λ µ ˆ µ × × × × × 8 4 2 3 3 ) to be exact equalities rather and and N ˆ λ 2.17 3 3 3 O , the scaling dimensions c 10 10 10 Pl Pl [ TeV] ˆ µ × × × M M UV . The smallness of neutrino masses is ) and ( and is taken to very high, of order the Planck M . Also shown are some derived parameters, in ˆ λ UV UV 2.13 UV M M M ), ( – 10 – 2.7 Λ [ MeV] c 9 400 8 40 2 4 400 2 05 40 25 400 2 . . . 2N . . ∆ , the Majorana mass term for the composite singlet neutrinos λ 8 4 9 3 4 2 N 85 3 25 3 . . . . . 2 ∆ 2 are order one at c 3 ˆ , and the cutoff scale µ 14 − GeV. The dimensionless couplings − Λ c 10 1 1 19 10 ˆ µ and × × 10 5 ˆ λ × 2 . 5 , we present a few benchmark sets of model parameters that lead to neutrino − 2 1 factors, this should suffice to give a sense of the current constraints on this class − 10 = 1 ˆ 1 1 1 1 2 1 1 4 λ 10 × Pl (1) . Some representative choices of parameters that lead to neutrino masses of parametrically 5 M O can lead to large changes in the values of the infrared parameters. It is instructive to In the first set of benchmarks the cutoff In table c I II V IV III 2N scale, small parameters necessary to realize therenormalization scale group seesaw evolution mechanism from at the lowIn scales Planck this arising scale scenario, from down even small to modifications theO to compositeness the scale scaling dimensions of the operators vanishes, the theory hasthe an exact SM lepton decouples number fromis symmetry. restored. the In the hidden Hence limit sectorthereby approximate that provide and symmetries an can the explanation explain for lepton the why number smallness either symmetry of neutrino of masses. the SM the parameters explained by theextent scaling to dimensions which the of SMoverall neutrinos the lepton mix number operators violation with respectively. their In compositeof the counterparts second neutrino and set masses the of benchmarks, is extent thein of explained, smallness the in ultraviolet. part, There by are approximate two symmetries global that symmetries can of play the a role theory here. In the limit that masses in the rightappear range. in We will the refer table.below to the the All electroweak benchmarks scale. the asdifferent benchmark The explanations I-V for benchmarks models in the fall the have smallness into order compositeness of two that neutrino scales broad they masses. categories that In that lie the provide first well set of benchmarks, measurements performed at energiesthe above low energy the theory, compositeness we will scalethan take eqs. to estimates. ( the Then, parametersup although in to the bounds andof projections models we and obtain the expected are reach only of accurate future experiments. outlined in the papermasses are is satisfied. fully In explainedinfrared. the by upper In the set the running of lowermasses of benchmarks, set of parameters the of the from smallness neutrinos benchmarks, the are of the explained, Planck the ultraviolet at scale neutrino cutoff least cutoff is in down lowered part, to to by 2 the symmetries. PeV, and the light Table 1 the right size. Shown are thethe dimensionless compositeness couplings scale particular the coupling constant and the resulting mass scale of the light neutrinos. For each set of parameters all the constraints JHEP03(2021)112 , even if UV , so that SM M UV M in the ultraviolet and by c that generates this mixing ˆ µ N O in the low scale models could is of order one in the ultraviolet, LH is very small at UV ˆ λ ˆ λ M to be low, of order the flavor scale, is a (slightly) relevant operator, so that UV c M 2N O , even if are small, so that both overall lepton number UV at the compositeness scale. Within this class of – 11 – ˆ λ M N resulting in low scale theories with the features we and /M c UV is instead a (slightly) irrelevant operator, overall lepton ˆ µ EW c M λv 2N O . Alternatively, a realistic spectrum of neutrino masses can be 1 that serves as an ultraviolet completion for the low scale models, and UV M TeV. There is now less room for renormalization group evolution to generate the 3 at the compositeness scale, and the size of SM lepton number violation, which is set in the ultraviolet and by In summary, our framework naturally allows for the generation of the small couplings It is important to note that the symmetries at In the second set of benchmarks we take 10 Λ ˆ λ / × c on the case in which the composite singlet neutrinos have masses at or below the weak scale. space to specific experimental observations. 3 Colliders In this section we discuss the phenomenology of this class of models at colliders. We focus This CFT could bepresent in broken the at table,are in approximate which symmetries. SM lepton number, hidden sector leptonrequired number, to or realize both, theenergy inverse parameters seesaw are mechanism possible. at low In scales, the and following a sections wide we range will of connect low the parameter and SM lepton number are only mildly violated atnaturally the emerge cutoff from dynamical scale, effects. asCFT in For above example, Benchmark one V. could easilyfor imagine which a the different scaling dimensions are such that these symmetries emerge at low energies. even if overall leptonBenchmark number III is in violated table obtained by if order overall lepton oneSM number in lepton is number the is a infrared, violated symmetryIV. by as to It order illustrated very is one good at in also that accuracy possible scale. at that This both is illustrated in Benchmark 2 small parameters necessary toscenario, accommodate symmetries a are low also scaleNeutrino expected seesaw masses model. to in play the Therefore, a right inlepton role range this number in can is be suppressing a obtained the symmetry if neutrino to masses. very good accuracy in the ultraviolet. This can happen I in the table.number can However, if also emergecase as it an will accidental also symmetryBenchmark at play II the a in the compositeness role table. scale, in in the which suppression of neutrino masses. This is illustrated in number must necessarily be ato symmetry naturally to occur very in good our accuracyis framework at since always low the irrelevant. energies. interaction This Forthe tends high mixing values can of easily bethe sufficiently light suppressed neutrinos. so This as remainsoverall to true lepton result even number in if is suitably violated small by masses order for one in the infrared, as illustrated in Benchmark µ by models, both these symmetries are violatedexpectation by order that one quantum at the gravity Planckconstraints violates scale, on in all line mixing global with between the symmetries. SM The neutrinos tight and experimental singlet neutrinos imply that SM lepton track the size of overall lepton number violation, which is set by JHEP03(2021)112 ). the the 2.3 (3.1) (3.2) ` gauge p bosons W Z − Z W p and and = . W ) ± N W p W p ( µ ) . ` ` p ` ( E v E 2 2 5 − − boson decays. γ W W − W m 1 m 3 µ ` W boson into a single flavor of lepton and γ E m represents the four-momentum of the de- N 2 N 3 / p W . The leading order matrix element contains p i − W 4 N 2 EW p ! ), v 2 2∆ 2 UV – 12 – / ˆ λ 3 2.3 M EW − 2 N 3 ˆ λv g ∆ i UV − = M 2
) N |M| p . Unparticle production from ( ¯ , as shown in figure u X 2 U 1 3 √ ig + gauge coupling while 2 the four-momentum of the charged lepton, and ± L ` ` Figure 4 = p decays → M Z ± i SU(2) W decays that give rise to multiple displaced vertices, and discuss the prospects of boson, and represents the energy of final state charged lepton. The partial decay width is W is the W ` W E g We first consider the inclusive decay of a given by, Here determined by integrating over the phase space densities of the charged lepton and the Here caying momentum carried away byspin the averaged matrix unparticles. element, after In summing over the the rest spins frame of the of final the state decaying particles, is unparticles, a single insertion of the interaction eq. ( Hidden sector statesthrough can diagrams involving therefore an off-shellhadronizes be neutrino. into final produced After states being consisting in produced,of of parameter the decays one space hidden or the of sector more compositeso composite singlet the the singlet neutrinos are neutrinos. search unstable is In on for much detector their timescales decay and products. the HL-LHC for these signals. 3.1 The hidden sector interacts with the SM through the neutrino portal interaction, eq. ( bosons into composite singletsearches neutrinos. for elementary We heavy neutral then leptonsspace translate that (HNLs) the is to currently existing determine allowed the constraints byin range the from the of data. future. parameter We also Finally, we determineinclude sketch the the reach of novel similar collider searches signatures of this class of models, which We begin with a concrete and detailed calculation of the decay rate of JHEP03(2021)112 - be W . and (3.8) (3.6) (3.7) (3.3) (3.4) (3.5) N N N p . ∆ 4 ! d 2 2 W 2 IR / , , to account 5 µ 2 m − / IR . , N 3 µ N ! . ). In the expres- ∆ ∆ → 2 W 2 IR ` can never actually ` 2.5 6 W
2 IR µ m N E E f 2 µ m 2 , 2 2 / ∆ / N . − 1 − W − + 2 ∆ − O 2 N m N W
W p Φ 4 W ∆ f m d 4 W = m 2 ` A m ) 3 . / 3 ` m 1 Φ 2 − 2 IR 2 IR . Physically, this divergence is − E d / 2 ), N 2 IR 2 µ µ ) 5 / N / 3 µ 1 − N ∆ 2∆ − − N p W − 2.23 A N ∆ 2 N 3 m ∆ − p W ! 6 IR , this expression must be considered as − ( ` Λ A µ N θ = p 2 W m we regulate this divergence. In practice, 2 IR 2 IR 3 ) µ ` µ − m 2∆ π − 0 N ) and ( 2 IR N 2 2 E p ( µ ) of the appendix. After rewriting the − W 2∆ θ 2.7 p ` W UV N ( 2 W W – 13 – ) = / E A.7 W m 4 1 UV M M m 2 (4) m ) IR − δ m π M N 4 for the total width is infrared divergent, with the − , µ ) 2 2 ∆ (2 2 | ` 2 | N 1 π λ π ` . Therefore the point 2 A ˆ λ | E α
C | (2 Λ 48 2 (∆ 2 N 2 = 2 W π f W EW U W captures the dependence on the scaling dimension 2 | ` m O / 2 m 96 W m ) 1 ˆ λv E Φ |M| g 2 | − 2 m 2 W 2 N W 2 π ∆ ) = g m , d /m Γ = × A U 96 ` d N 2 ` 2 IR , the phase space volume factor, was given in eq. ( p =1 / = 3 N X n α 3 d , µ ` → A ` → N Γ lim N 1 E d the integral over dE W ∆ 2 ) = (∆ 2 3 f U Γ( ) / ` 5 1 π (2 → < = W N ` ∆ Γ( Φ . Its detailed form is given in eq. ( d IR Since the final result isjust sensitive an to estimate. the As value of a consistency check we consider the limit that partial width as a functionconventional HNL of mixing the angle infrared using parameters, eqs. it ( can be expressed in terms of the The function µ of order the compositenessbe scale reached. By choosingthere a is an nonzero additional value cutoff of sufficiently arising energetic from to the be requirement observedgiven that in by, the the decay collider products environment. of The partial width is then For divergence arising from theregulated neighborhood by the of fact that thein conformal the symmetry unparticle is sector, broken the and composite so the singlet final neutrinos, state are particles not massless but have masses of the final state lepton, The expression for sion for the unparticle phasefor the space fact we that have conformal introducedintegration invariance an is over infrared broken the at cutoff, angular the variables compositeness we scale. obtain Performing an the expression for the energy distribution The phase spaces of the charged lepton and the unparticles are given by unparticles, JHEP03(2021)112 0 0. for 20 10 , for ` 10. 2 | E ` 15 -1 1000 α N 10 U | 10 α -2 -1. 10 P ) =4GeV N .4Gev) =0.04 10. 100 N 5 -3 W W 10 /m /m l l -4 1-2E 2 N 2 N 1-2E 10 2.4 =m -2. =m 4GeV 4 = 10 . 00 GeV 0.04 = 2 IR N 2 IR -5 μ N μ N m 10. m 10 ∆ (m produced N max (m produced N max -6 0.04 GeV 1 . The solid, dashed, and dotted lines 2.2 10 N 1 ∆ -3. -7 4 2 0 8 6 4 2 0 10 -2 -4 -2 -4 -6 10.
10 10 10 10 10 10 10 10
10 10 10 10 10
N
) / ( / Γ Γ / ) / ( / Γ Γ / 2 d d 1 2 d d 1 W L W L m m E Δ E 0.4 GeV 2.0 of 7/4, 2, and 9/4, respectively. The green, red, as a function of the charged lepton energy 0. , normalized by the IR mixing angle 1.0 – 14 – N U 10. U ` ∆ ` 2 N 4 GeV value and show the distribution in log-log scale. → 1.8 → 100 m = ` -1. = N E 0.8 W 2 IR m W 10. 50 μ and scaling dimension 7/4 = N . Gev) =0.4 -2. N 1.6 N Δ 8/4 9/4 W m 0.6 W 10. /m 150. /m l l , as a function of the scaling dimension 10 produced /0.4GeV) N -3. N 1-2E 2 N
1-2E
0.06 0.04 0.02 0.00 0.10 0.08 0.4
m
ℓ N GeV 4 = )/| →ℓ ( W Br
U 10.
=m N 2 GeV 0.4 = 2 IR . GeV 0.4 GeV 0.04 m N μ m -4. 100. a N×( m max 0.2 (m produced N max 10. 1 . The differential width of . The branching fraction of 50. -5. 6 4 2 0 2 0 0.0 -2 -4 -6 -2 -4 10.
10 10 10 10 10 10
10 10 10 10 10
) / ( / Γ Γ / ) / ( / Γ Γ / 2 d d 1 2 d d 1 W L W L m E m E various benchmark choices of represent benchmark scaling dimensionand blue of curves correspondsthe to upper singlet axis neutrino ofkinematics. masses the The of upper figure 0.04, left wepanels 0.4, panel also zoom shows and in the show 4 GeV, overall to the distribution respectively. near maximum in the On log-linear number maximum scale, of while singlet the rest neutrinos of allowed by Figure 6 Figure 5 benchmark choices of JHEP03(2021)112 , , 4 / and 9 nN (3.9) , and , (3.10) 2 4 + that is . / 0 ± and , W ` N 4 m 04 / . → 8 ' in this section , ± 4 ` = 0 N / . The results are E W ˆ λ m N , is dominated by for various bench- N = 7 M = . N -boson center of mass ∆ N ` ∆ IR ` W E ∆ µ E separately, but now with 2 2 N − , the greater the fraction of , which is plotted on a linear − , normalized by the infrared ` Z M in the U IR Z E ` ` µ m bosons to decay to unparticles. m ) reproduces the standard result 3 E . ` → Z ` 3.7 . In the rest of the panels, we show W W in these three panels, reflecting the E E m N to final states that include a charged ` 2 m W 3 ∆ E − W − N 2 EW , eq. ( 1 v N 2∆ 2 UV s ˆ λ m M since it is simply a delta function. In the top N W ) = – 15 – 2 2 m m 0 / g IR = and the decrease with multi-body phase space of 6 µ + = 3 ), 2 λ g max N ( 3.2 /C n ∆ 1 = 2 bosons strongly favor low numbers of singlet neutrinos in the . The multi-body phase space dominates the behavior and ) |M| 2 W W , as a function of the scaling dimension 2 X | decay, eq. ( /m ` . Then, setting . This branching fraction is sensitive to the choice of the infrared 1 3 α 1 boson decay into hidden sector states can be calculated in the same 2 IR N N W decay. The square of the resulting matrix element takes a form very Z U , µ → M | N λ W we show the branching fraction of P C (∆ 5 f we show the differential width of the × 6 an odd integer. The maximum number of final state singlet neutrinos 2 / 1 n − The rate for So far we have discussed the inclusive rate for In figure plotted on a logarithmic scale, zooming into the threshold region. From this figure we N ∆ ` GeV. We omitted the case of manner as for similar to that for the unitarity of theneutrinos, area or under equivalently the theevents larger with differential charged the curve, lepton infrared the energiesthe cut at heavier unparticle lower off the decays values. of composite Duefinal to singlet state. the shape of this distribution, E can see that the chargedis lepton separated energy from favors the itthe kinematic only threshold horizontal by of axis the extends non-vanishingdifferent to mass benchmark of different composite the values singlet of HNL. neutrino Note masses. that As the a range result of of this behavior and and for three benchmark4 masses of the compositeleft panel, singlet we neutrinos, show thescale, overall distribution for as the a benchmark functionthe values of of same the distribution masses for and each of the benchmark values of In figure lepton and an arbitraryenergy. number The of form of singlet thesepresented neutrinos distributions for as is three independent a of benchmark function the values of parameter of the the charged scaling lepton dimension, “hadronize” into singlet neutrinoswith so the ultimatekinematically fate allowed, of for a the given decayframe, charged lepton is is energy given by, cutoff. Thetwo dependence effects, on the scalingA increase dimensions, from as adrives function the decrease of of the branching fractions into unparticleHowever, states. below the scale at which conformal symmetry is broken the unparticles must for an elementary HNL, aswhen expected. presenting Going results. forward, we shall set mixing angle mark choices of In this limit, JHEP03(2021)112 . In . (3.12) (3.11) of the 8 N 2 Z 2 IR m m µ , N ∆ . The constraints f 7 2 / 1 ], shown as the brown − N , and present the bound 54 ∆ , N A 53 M 3 − = N α 2∆ N . M Z 2 | Λ e m as a function of the mass α N 2 Ne U boson decays [ | U Z N N bosons into elementary HNLs that mix with =1 W N X boson decay shown in the figure are based on α m M Z – 16 – ≡ Z 2 2 | | ` are degenerate, and α and Ne α 2 N λ boson decays searched for by the DELPHI experiment U W U | N C | Z W 2 ) 2 π 0 g 192 + boson is then given by, 2 Z g ( Z m N =1 N X α ] are shown in blue. The LEP search covers a broad range of signature 55 . The limits from )= N U ` ∆ : above a few GeV the constraints come from the current LHC searches for : LEP dominates the current constraints in the 2-40 GeV regime. The bounds on ¯ ν ]. Although the resulting limits will be stronger than the actual bounds, the 52 → prompt decays, since theweaker constraints than from those long-liveddecays from at particle a beam searches combination dumps. at of LEP prompt are searches at LEP and D-meson LEP HNL production from rare at LEP [ spaces, including HNLs givingdeposition rise in to the prompt decays, calorimeters. displaced vertices In and our energy analysis, we consider only the bounds from LHC promptly decaying HNLs produced in curves. Z The future reach of HNL searches for this class of models is shown in figure The current constraints on this class of models are displayed in figure In our analysis, we consider only the bounds coming from final states that include • • Γ( the figure the current limits in the composite neutrino mass versus mixing angle plane are the following searches: are expressed ascomposite limits singlet on neutrino. the Thedimension mixing bounds are angle shown for four different choices of the scaling electrons, which tend tosimplicity be we assume the that strongest allin in the terms flavor-democratic of models. the effective Furthermore, mixing for angle squared, the rates for unparticle productionHNLs to the [ corresponding rates fordifferences production are of expected elementary tothat be decays modest into since states the with results very of few the composite previous singlet subsection neutrino indicate are heavily favored. In this subsection weexisting determine searches the for current thethe bounds decay SM on of neutrinos. this classOur We results of also are models obtained discuss by under thein the recasting expected assumption the that production reach all of of decays into just similar the a searches hidden single sector in result composite the singlet future. neutrino, which allows us to compare 3.2 Current constraints and future reach from HNL searches The partial width of the JHEP03(2021)112 2 2 2 2 eγ 10 10 10 10 → µ eN π→ π→eN π→eN π→eN D→eN D→eN D→eN D→eN 1 1 1 1 2 N 2 N 2 N
10 10 10 10 m m m = = = 2 IR 2 IR 2 IR conversion and K→eN K→eN K→eN K→eN mixing) mixing) mixing) e μ-e μ-e μ-e → µ Z→νN Z→νN Z→νN Z→νN μ μ μ (maximal (maximal (maximal 0 0 0 0 10 10 10 10 μ→e μ→e μ→e & & & W→lN W→lN W→lN W→lN mixing) -e μ (maximal e μ→ & μ→eγ μ→eγ μ→eγ μ→eγ (GeV) (GeV) (GeV) (GeV) N N N N m m m m – 17 – -1 -1 -1 -1 10 10 10 10 -2 -2 -2 -2 10 10 10 10 2 9/4 7/4 = = = 3/2 = N N N N Δ Δ Δ Δ . The existing limits on the IR mixing angle as a function of composite neutrino mass , the compositeness scale) for final states involving electrons. These bounds are a rein- Λ -3 -3 -3 -3 0 0 0 0 10 10 10 10 ∼ -4 -6 -8 -2 -4 -6 -8 -2 -4 -6 -8 -2 -2 -4 -6 -8 -10 -12 -10 -12 -10 -12 -12 -10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10
10 10 10 10 10 10 10 10
N
Ne Ne Ne Ne | | | |
U U U U
2 2 2 2 m terpretation of constraints onregions HNLs, correspond as to discussedmeasurements the in in flavor the the dependent text. maximal constraints electron-muonsection. The flavor from mixing lower the scenario, and which upper is discussed shaded in gray the next Figure 7 ( JHEP03(2021)112 2 2 2 2 10 10 10 10 projection μ→e projection μ→e projection μ→e projection μ→e 1 1 1 1 10 10 10 10 are shaded in gray. Details of the xsigLimits Existing Limits Existing xsigLimits Existing xsigLimits Existing 7 Codex-B Codex-B Codex-B Codex-B plane, for different benchmark choices of 2 | FASER2 FASER2 FASER2 FASER2 SHiP SHiP SHiP SHiP Ne (GeV) (GeV) (GeV) (GeV) U N N N N | m m m m – 18 – DUNE DUNE DUNE DUNE FCC-ee FCC-ee FCC-ee FCC-ee 0 0 0 0 10 10 10 10 H LLP LHC MATHUSLA LLP LHC MATHUSLA H LLP LHC MATHUSLA H LLP LHC MATHUSLA 2 = -mixing angle squared . The current limits detailed in figure 9/4 = 7/4 = 3/2 = N N N N N Δ N Δ Δ Δ ∆ m conversion experiments are in light gray shade and dot-dashed line, respectively. e - µ . Future projections for constraints on the composite neutrino parameter space, in the -1 -1 -1 -1 0 0 0 0 10 10 10 10 -2 -4 -6 -8 -2 -4 -6 -8 -2 -4 -6 -8 -2 -4 -6 -8 -10 -12 -14 -12 -14 -10 -12 -14 -10 -12 -14 -10 10 10 10 10
10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10
10 10 10 10 10 10 10 10 10 10 10 10
Ne Ne Ne Ne | | | |
U U U U
2 2 2 2 singlet neutrino mass the scaling dimensions future projections can betions found from in the the text. The flavor dependent current limit and future projec- Figure 8 JHEP03(2021)112 – ]. for ] to 54 5 59 , . The 63 − (3.13) ]. The 8 . 58 10 -mediated 56 2 Z ≈ τ and κ W bosons can give rise , based on a long-lived . W -boson decays [ 8 as the blue curves, using W 8 meter , are especially promising. In − ` c µ + keV boson mass is hierarchically larger + ` 1 , the number of kinematically allowed W − i + ! ν 2 ν | ` For a composite singlet neutrino mass of → 01eV m α . ]. N 0 1 . N U 60 | Z , as well as the number of appropriately weighted – 19 – 5 3 5 N and corresponds to a benchmark value of π W , multiple singlet neutrinos can be produced in the m c N is an odd number. These composite singlet neutrinos Λ 2 F -mediated process also contributes to 192 bosons over its lifetime, it is expected to have excellent Z /µ G n M GeV ν ` κ W -pole run [ m W ` × Z . In our analysis we use the results presented in ref. [ 12 ∼ c m X . The µ bosons decays shown in the figure are based on the following 2 10 60 , where | ` 25 . Z α ≈ ∼ 8 N and n N α U ≈ ν | + and cτ µ ± m ` κ W counts, for a specific lepton flavor , it is necessary to take into account interference between the ≈ i → i κ e κ : the projected HL-LHC reach is shown in figure ± ]. New search ideas at the HL-LHC will be able to further improve the cov- κ ]. Therefore, in this class of models, the decays of : projections for FCC-ee are shown in orange in figure W 57 62 , GeV, 61 , particle analysis from its nal, based on theCMS calibration [ data for theerage. displaced For dilepton a analysis more carried optimistic out projection by withoutFCC-ee background, see, e.g., refs. [ HL-LHC a conservative analysis of long-livedanalysis particles employs produced in a dilepton trigger with an additional displaced lepton as the sig- = 1 The lifetime of the composite singlet neutrinos is given approximately given by, In calculating 58 • • 1 , N the above choices of model the details ofof threshold the effects, composite finite fermion singlet mass neutrinos. corrections, etc., on thechannels lifetime when considering final states involving charged leptons. The parameter decay channels mediated by the off-shell decay channels mediated byM the off-shell mixing angle scales as 56 to highly characteristic signalsexpected to involving produce of multiple order displacedreach for vertices. this class Since of models the in LHC much is of parameter space. same decay, are not stable, but decay toto SM particles final through states the that weak interactions. containcertain charged At regions colliders, leptons, decays of parameterdisplaced space the vertex singlet signatures neutrinos that are have long-lived, not leading been to constrained striking in previous experiments [ 3.3 Novel signatures This class of theoriesmodels can with give an rise elementarythan to singlet the exotic neutrino. compositeness signals If scale, that the are not present in conventional future limits from proposed searches: shaded in gray, with the flavor-dependent constraints in a lighter shade. The projected JHEP03(2021)112 . at 0 0 (3.14) 10 10 LHC cτ ) R 7/4 9/4 = = max N N n Δ Δ to determine (¯ -2 -2 LHC β 7 ) 10 10 R max 2 2 n l l (¯ decay, we consider two N N γ U U -4 -4 | | =1) =1) − (n (n W 10 10 max max =n =n n n 4 GeV 4 GeV exp = = N N m 0.4 GeV 0.04 GeV m 0.4 GeV 0.04 GeV − -6 -6 (Dotted) (Dotted) 1 or one singlet neutrino, respectively. 10 10 Solid Solid × max ) 7 4 1 7 4 1 10 10 n N 10 10 10 10 10 10
10 10
∆ - # - # LHC LHC HL within N of fNwti HL within N of . The solid or dashed lines show how the results . The kinks in the curves correspond to the , 1 N N 0 0 ), averaged over the differential distribution in ∆ m 10 10 – 20 – ( 3.9 2 3/2 , we can place an approximate upper limit on the meters at the 13 TeV LHC with an integrated lumi- max = = N n 6 N Δ Δ × -2 -2 = 10 ) we show the total number of composite neutrino decays 10 10 IR 9 LHC , µ 2 2 boson to the hidden sector after averaging over the kinematics R l l N N N defined in eq. ( ∆ U U -4 -4 | | =1) tot W W , (n Γ 10 10 N and scaling dimensions max max =n M n N n . This figure should be considered together with figure 4 GeV 4 GeV ( 1 = = m N N decay is simply taken to be − `N W m 0.4 GeV 0.04 GeV m 0.4 GeV 0.04 GeV Γ -6 -6 (Dotted) W 10 10 × Solid represents the maximum number of composite singlet neutrinos that can be pro- W 3000 fb 1 7 4 1 7 4 n . The expected number of composite neutrinos decays at the HL-LHC for various bench- 10 10
). In the second of the two limiting cases, the number of composite singlet neutrinos 10 10 10 10 10 10
max . 10 10
- # - # LHC HL within N of LHC fNwti HL within N of ¯ n i 3.5 Given the large proper lifetime of the composite singlet neutrinos and their energy N n h sult on the number oflimiting composite cases. singlet In neutrinos the produced firstically per limiting allowed case, value this number iseq. taken ( to be the maximallyproduced kinemat- per of individual events. Inwithin a figure detector radius of nosity of the regions of parameterfor space three that different benchmark are values not4 of GeV, already the shown composite excluded. in singlet green, The neutrino mass, red results 0.04, and are 0.4, blue presented and respectively. To explore the dependence of the re- Here duced in the decay of a distribution as indicated fromtotal figure number of composite singletthe neutrino HL-LHC decay as, events within a detector radius Figure 9 mark choices of the vary by assuming each unparticle produced results in JHEP03(2021)112 . as N` 7 , 2 U | (4.1) (4.2) ! e 2 m α 2 IR N m µ U , | N N =1 ∆ N α
and the infrared g P N × ≡ 2 ∆ 2 / ). As before, we can | 1 − Ne , N ` U A.14 ∆ | E A m 3 m − , the scaling dimension. With N 2 N ], where the shower particles are ! 2∆ ∆ 2 69 / – 3 − m 64 UV EW N m ∆ UV M ˆ λv M 2
| ` from rare meson decays are shown in figure 2 m α boson decays into composite singlet neutrinos f – 21 – N 2 N | Z U 0 | . This shows that the HL-LHC can indeed probe m 2 5 qq | 0 − V λ | , the square of the spin averaged matrix element is and qq 2 F 10 C U V 2 | ` G decay, which could be an crucial consistency check for the π W 2 N to 32 m → 7 W = 4 2 m − 2 f m 10 2 F G M| | m m captures the dependence on the scaling dimension N =1 N ) X α 2 W 2 IR ). µ m ]. ) = , U 52 N 3.12 . The details may be found in the appendix in eq. ( is the meson decay constant. The corresponding partial width is then, ` IR (∆ m in the range from → g f µ 2 | m The current bounds on the effective mixing angle squared As with the calculation of For the meson decay N` Γ( U neutrinos [ a function of compositeThe neutrino mass results are presentedthe for exception four of different the choices pion of decay bound, the charged meson searches look for the decay detailed in the previous section,into we constraints translate on the the existingunder parameter constraints the on space assumption elementary of that HNLs our allsinglet model. decays neutrino. to As the This before, hiddenson sector allows our result of the limits in the bound are just rate obtained to a for single be composite unparticle obtained production from to a the straightforward rate compari- for production of elementary sterile where cutoff express the partial widthusing in eq. terms ( of the conventional sterile neutrino mixing angle where of future beamCodex-B dump and experiments MATHUSLA. and the proposed LHC-basedgiven experiments by, FASER2, 4 Beam dumps For compositeness scales belowconstitute a a few powerful GeV, the probe rarebounds of decays on this of this class scenario pions, of from kaons models. the and rare In B-mesons decays this of section mesons. we In discuss addition, the we current explore the reach often hidden pions or hiddenparticular, photons. in Instead, lepton(charged our and modelticle neutral)-rich predicts calculation final an allows HNL states. for shower, an anddifferential Here, approximate cross in in prediction section addition, from of the theunderlying accompanying unpar- theory charged and lepton dynamics. (smaller mixing angles). Wedecays see within that the in general, LHC| if for we discovery, require the about HL-LHCunexplored 100 can regions singlet probe of neutrinos values theneutrino of parameter is the closely space. mixing related angle to Note that that of the dark phenomenology showers of [ unparticle transition between prompt signals (larger mixing angles) and collider metastable signals JHEP03(2021)112 . 8 . ] experiment at 7 71 ]. The highly intense 77 -mesons, kaons and pions, all of D ]. These are shown as the orange curves Proton-on-Target (PoT). It is sensitive to 70 18 – 22 – 10 . ] lead the constraints in the 130-400 MeV regime. × ]. 7 72 ], which has a higher beam energy than other beam 60 74 . 76 1 ] experiment searched for long-lived HNL decays from a high PoT can copiously produce 74 22 10 ]. The resulting bounds are shown in green in figure × 73 3 ] also searched for HNLs from kaon decays. The resulting limits are weaker than those 2 . We also include the bounds from the recent PIENU [ . . : the strongest constraints below the pion mass threshold are from searches 75 : searches for HNLs from D-meson decays were conducted at the CHARM we show the projected reach of future experiments. In this plot, the existing 7 8 7 : the DUNE near detector can search for the decays of long-lived particles 8 : searches for charged kaon decays into a charged lepton plus a HNL performed : the NA62 [ : the SHiP experiment [ beam with which can act asbelow sources the for kaon singlet mass neutrinos. threshold than DUNE other has proposals, greater as can sensitivity be reach seen from figure to 5 GeV from thein decays figure of B-mesons. The sensitivity is shownDUNE as the green curves produced from the beam dump used to produce neutrinos [ SHiP dump proposals, can probe long-lived HNL decays. It is sensitive to HNL masses up HNLs produced in kaonin decays, figure leading to the limits shown as the lower red curves CHARM experiment [ NA62 intensity beam dump with the limits are provided. PS191 at CERN’s PS191 experimentThese [ are shown in red in figure TRIUMF at TRIUMF for softpion positrons exclusive from decay positively ratein measurement charged figure [ pion decays andTRIUMF the shown in overall the same color which sets a stronger constraint in regions where In figure T2K ND280 [ • • • • • • 2 from NA62 and we do not explicitly include them here. sensitivities to composite singlet neutrinosfuture produced from experiments, meson including decays SHiP, in Codex-B,of several different the MATHUSLA, DUNE projections and are based FASER-2. on Most ref. [ limits on the parameter space are shaded in gray. Included in the figure are the projected For concreteness, in ourstates involving analysis electrons. we Since havescalar the only meson heavy decays, neutrino considered and provides constraints the theto electron arising chirality be is flip from the the required lightest final for strongest chargedfollowing lepton, in searches these flavor-democratic for constraints elementary models. tend HNLs. The limits we present are based on the products of elementary HNLs into final states involving charged leptons or charged pions. , ↵ 49 (6.3) (6.1) (6.2) ]. The 53 , . the spectral anomaly [ ) 52 , 2 µ 2 ↵ r ✏ ( 4 i O 2 µ ⇠ + r + 2 ). Expanding in the / p + from a heavy neutral µ 2 f i 1) M 2 f r e
JHEP03(2021)112,r r f 3 ] and cut o r log 2 2 µ ( ! r p 6 55 f x I r 2 µ + / 5 as 2 f )andanunparticlepropagator. r x N ( +18 , . ↵ x 2or 8 f 2.3 +2 49 r eγ . The transitions 2 µ conversions will also be probed with ]. The (6.3) (6.1) (6.2) 2 IR r 4
2 + N 2 IR 1 − e 2 k → 3 µ α ∆ 2 z ]. It therefore places a constraint N z µ M A − 6 − ( 6 U 5 2 85 2 − µ − − 2 N 2 α N 5 M M ∗ N z 1 4∆ M N 2∆ dM U − 2 2 2 M . Again, expanding in the small parameter + 3 N 2 IR ∞ =1 k ) N µ W 7 M N α dM Z z W . Using the CLs method and requiring that M M 2 P 2 2
2 IR 10 ∞ / 1 " µ 1 /M − – 25 – M 2 ! Z − µ ! π / 4 λ 5 N 1 2 10 2 2 = | − C ∆ − , m µ ) × 2 N λ 3 N α A α 2 (0 ∆ π ∆ C N 3 2 ) 7) I . . Unfortunately, this is of the wrong sign to explain the M A U | 7 2 − IR N 2
128 ) z − / 1
± 3 m 2 × W 3 . This translates into limits on the mixing angle − − k 3 2 µ . π ( 2 ∼ N 10 . 2 z m 2 /M arises from a diagrams of the same form as in figure ∆ λv − ( Λ µ k F √ ) = Λ 11 10 λ G dz eγ 16 eγ = (31 , m C × and ∞ IR 5 N µ W z =1 → ]. In this subsection we determine the hidden sector contributions to → a N X α Z 2 W ∼ < 2 µ µ µ ∆ π | ( × − 82 2 , m µ U U /M M/M a F √ ( | 81 = IR I G 16 BR µ µ U ≡ ). The insertion of the mixing angles and the additional propagators makes the a leads to a tractable result = = I 5.2 U µ ∆ W IR a z The process /M µ experiments [ these processes. resulting amplitude can beto evaluated the in branching the ratio exactly given same by, way, resulting in a contribution 5.2 Lepton flavor violation In general, the couplingsbe of flavor the aligned hiddencharged and leptons. sector can The to therefore lepton theprocess flavor give SM will violating rise be muon neutrinos probed decay to with are greatly flavor not increased violating precision expected at processes the to upcoming involving COMET and the Mu2e the composite neutrino modelcompared hypothesis to not the be SM-only excludedcontribution, hypothesis, at we the are 95% ableas confidence to the level place red as lines a in bound figure on the hidden sector with discrepancy in the muon anomalouson magnetic moment the [ parameter spacenetic of moment the is model. The observed excess in the muon anomalous mag- m diagrams, where loop integral more complicated. However, taking advantage of the identities the unparticle diagram can be evaluated as an integral over the difference of two simpler in eq. ( JHEP03(2021)112 . 1 (3.1) | , e ) . p N e 1 ( µ U (5.12) | u . The (3.1) i j ) ] and is n U , R e ) capt P 87 Z p µ Γ ( W µ m u is "local" (i.e. it only e + i i j n ) n L µe R P F , e P Z (c) Z Penguin Diagram 4 µ . W ] places a constraint m 2 W u, d u, d 1 10 | ( ective Lagrangian relevant for . µ m d 2 ff conversion. They are as a bound on 86 µe | is "local" (i.e. it only ) is violated. (3.1) e F i + i j G n ) 7 u e n , ⌫ L conversion in the type-I seesaw model. e ) µe N e q Z e p ⌫ P e 2.18 − 3 ( F U e µ propagator of the photon between the | (c) Z Penguin Diagram µ . The second term in this equation - − W 0 10 to (e) Box Diagram µ 2 W i u m p g-2 u, d u, d i ( j = µ ective Lagrangian relevant for A µ ) /q n W μ ff 1 -penguin diagrams. As before, µe R 2 =3/2 =7/4 =2 =9/4 L
i µ dd p N N N N P Z j G U Z e P n Z coupling- is the only one contributing for an u µ Δ Δ Δ Δ ) ⌫ α i + (2 W 2 conversion in the type-I seesaw model. µ = q n m e N –5– µe ⌫ u e and muon capture rate 10 is "local" (i.e. it only qq q e µ + i propagator of the photon between the U G e F U . The second term in this equation - n W 0 µ ) 6 to (e) Box Diagram L 2 µe A i p α P µ W Z µ F e ects. The W and Z mediated diagrams are obviously all ∗ /q W N (c) Z Penguin Diagram 2 ff (b) Z Penguin Diagram 1 W m conversion (blue lines) experimental measure- q + 1 U u, d u, d L ( ective Lagrangian relevant for µ dd ( p u, d u, d µ (GeV) transitions with unparticle neutrino particles in e P µ Z ff 10 coupling- is the only one contributing for an µe - A ) µe N N i =1 i e j ( G µ = µ i n n m u F | j N α , has been carefully calculated in [ ⌫ n –5– µe d h conversion in the type-I seesaw model. e q 2 qq e ) q → 0 – 26 – . This is shown in figure ⌫ G e e e μ→eγ P p ! → µ 2
WW propagator of the photon between the ]). One can therefore divide the e 6 ( . The second term in this equation - µ
W ) µ 0 0 to (e) Box Diagram e 2 e i p µ p (d) Box Diagram 25 W u ( 10 α R µ ects. The W and Z mediated diagrams are obviously all ) Z /q -box, penguin and W 2 q N ff W (b) Z Penguin Diagram 1 V ( L q µ dd p µ U ( , mass number W u, d u, d P µ uu Z coupling- is the only one contributing for an
✏ of 3/2 (solid lines), 7/4 (dashed lines), 2 (dotted lines), 9/4 µ e ) µ i Z µe µ = U n α i 2 W N –5– µe n μ→ i 2 W F ∗ j -shell photon exchange and it involves 2 powers of the photon momentum in -1 U N qq q n d M G h e ∆ ff α 2 W capt 2 10 ) U mediated diagrams contribute to both classes of transitions, extended and local. 6 matrix element can be written as 0 ) e conversion rates 2 F Γ p µ ⇡ WW W ieg ]). One can therefore divide the e 2 ( (green lines), and N ects. The W and Z mediated diagrams are obviously all e W G =1 e e 2 denotes the photon momentum, . The five classes of diagrams contributing to ff π (b) Z Penguin Diagram N α 2 (d) Box Diagram 25 2(4 q u conversion is induced by a series of gauge boson mediated diagrams given in Fig. ( u, d u, d ) q eγ ! to µ u, d u, d P q -2 W (a) Photon Penguin Diagram = e µ = µe
µ ( For a rigorous calculation of the rate it is necessary to separate the local contributions from µ 0 i 10 -1 -2 -3 -4 -5 F µ → j e n µ uu d to ✏ h 10 M
10 10 10 10 10
)
i
→ µ 0 μ N
| e
U Figure 1 i where mediated by the photon-leptonon-shell "dipole" photon and isaccounts non for o local, whereasthe the numerator which "monopole" compensate term lepton the long and range nuclei lines [ are sensitive to atomic electriclocal. field e The The 3 3.1 Calculation of theIn the rates type-I seesaw framework, violationµ of charged lepton number arises atThe the various one loop contributions level. to thetransferred process by the can be photon,penguin divided by diagrams, in the whereas those Z the in bosonquark latter which case, or the the processes via internal momentum corresponds fermions two is mixings, in to W the in a bosons. loop order box must to The have diagram. avoid non-degenerate first a masses Alike two GIM and to proceed non cancellation. trivial the via the "extended" ones. This stems from the fact that extended contributions, unlike local ones, 2 W 2 p µ WW . They include ]). One can therefore divide the e n ( -shell photon exchange and it involves 2 powers of the photon momentum in e R M (d) Box Diagram 25 ff 2 u W 2 ) mediated diagrams contribute to both classes of transitions, extended and local. 12 matrix element can be written as ) conversion rates q W ( ⇡ ieg µ e W µ uu e ✏ denotes the photon momentum, . The five classes of diagrams contributing to 2(4 i 2 W conversion is induced by a series of gauge boson mediated diagrams given in Fig. U q n . Constraints on neutrino mixing as a function of the singlet neutrino mass scale from ! to . Loop diagrams contributing to -shell photon exchange and it involves 2 powers of the photon momentum in u, d u, d M (a) Photon Penguin Diagram = e µ ff 2 W µ 2 For a rigorous calculation of the rate it is necessary to separate the local contributions from µ mediated diagrams contribute to both classes of transitions, extended and local. matrix element can be written as ) conversion rates ⇡ to ieg M e W i e denotes the photon momentum, . The five classes of diagrams contributing to There are several one loop diagrams that contribute to Figure 1 accounts for o the numerator which compensatelepton the long and range nuclei lines [ where mediated by the photon-leptonon-shell "dipole" photon and is non local, whereas the "monopole" term the "extended" ones. This stemsare from sensitive to the atomic fact electric thatlocal. field extended e contributions, The unlike local ones, The µ The various contributions to thetransferred process by the can be photon,penguin divided by diagrams, in the whereas those Z the in bosonquark latter which case, or the the processes via internal momentum corresponds fermions two is mixings, in to W the in a bosons. loop order box must to The have diagram. avoid non-degenerate first a masses Alike two GIM and to proceed non cancellation. trivial the via 3 3.1 Calculation of theIn the rates type-I seesaw framework, violation of charged lepton number arises at the one loop level. 2(4 conversion is induced by a series of gauge boson mediated diagrams given in Fig. q ! to u, d u, d (a) Photon Penguin Diagram = e µ µ For a rigorous calculation of the rate it is necessary to separate the local contributions from µ propagating in thegiven loop. by, This rate, for a nucleus of atomic number alone, under the simplifying assumption shown in figure we work in thediagrams can insertion be approximation. related to In the corresponding this amplitudes limit for the the case amplitudes of of elementary HNLs the individual the loop. The current limit on this branching ratioon from the the matrix MEG experiment product [ Figure 12 Figure 11 muon g-2 (red lines), ments, for benchmark choices(dot-dashed of lines). In the gray region the boundary condition in eq. ( to M i Figure 1 where mediated by the photon-leptonon-shell "dipole" photon and isaccounts non for o local, whereasthe the numerator which "monopole" compensate term lepton the long and range nuclei lines [ are sensitive to atomic electriclocal. field e The The The various contributions to thetransferred process by the can be photon,penguin divided by diagrams, in the whereas those Z the in bosonquark latter which case, or the the processes via internal momentum corresponds fermions two is mixings, in to W the in a bosons. loop order box must to The have diagram. avoid non-degenerate first a masses Alike two GIM and to proceed non cancellation. trivial the via the "extended" ones. This stems from the fact that extended contributions, unlike local ones, 3 3.1 Calculation of theIn the rates type-I seesaw framework, violationµ of charged lepton number arises at the one loop level. JHEP03(2021)112 e ], - µ 88 con- . In ], are (5.13) e - conver- 87 and ] to the µ e (90% CL) eγ 89 , 13 − 70 GeV and − → µ 82 ∼ > 10 µ eγ N × , given in [ 7 → M u,d 3 , < F − µ , under the simplifying ) e | N x , under the assumption ( → 2∆ Au µ N µ u,d N e ] and Mu2e [ R U F U | W 81 2 Λ / 5 M − N , and the rate for muon conversion 2 1 ∆ , the spectrum of light states in the ) Λ λ . The functions ) IR C p ≈ − ( x x ) N W V − Z M M x 2 ( + comes from the SINDRUM II experiment [ √ e A below 50 GeV, showing the complementarity of dx – 27 – ( → . We can see from the figure the complementarity / e N µ IR ) ∞ 2 α | x R A M N Z 2 U N µ 1 2 we show the current constraints from µ − U − α | π N ∗ Z N 11 2 ∆ = U A 2 . We see that in the limit of large mixing, = (
N 2 =1 e | d N X × α α N /F N e → U u . Cancellations between the neutron and proton contributions to we present the current and future constraints on this class of U µ | 2 F . In figure ) α 8 u,d ∗ N . These bounds are prsented as limits on 16 = W F − U N 2
10 /M M N and e =1 α N α IR ∼ < 7 N µ P e U
µ → α = ( Al µ ∗ N R U IR x N In figures Currently, the strongest bound on =1 N α P beta decay. At scales belowneutrino the sector compositeness consists scale ofsmall Majorana just masses the from the SMmomentum nonrenormalizable scale neutrinos Weinberg in operator. and neutrinoless The double antineutrinos,to characteristic beta which the decay spacing now is between of possess nucleons order in 100 MeV, the which nucleus. corresponds Since the Weinberg operator is the lowest these different types of searches.different experiments In may the help event uncover of the a nature discovery, of multiple the measurements underlying from 6 theory. Neutrinoless double betaIn decay this section we consider the implications of this class of models for neutrinoless double sion currently gives thethe strongest future, bound assuming large onlimits mixing, this for Mu2e class will compositeness of provide scalesfuture models greater above experiments sensitivity for a than above the few 50pected GeV. current GeV to and Future have greater will collider sensitivity have and for the beam greatest dump reach experiments of are all ex- between these two types of lepton flavor violating experiments. models, with the limitsversion from included. the lepton The flavor bounds violating processes are presented as limits on but is expected tolevel of be considerably improvedconversion by in COMET green [ andneutrino blue masses lines, respectively, overassumption a broad range of composite singlet with the rate can stillrelative occur to in the the case case of of elementary composite singlet neutrinos, neutrinos. butwhich at is based masses on that measurements are on shifted gold. This limit stands at form factors determined byHNL. the There one is a loop value diagrams ofapproximately the cancel, and neutrino are mass for a whichis function the suppressed, neutron of resulting and the proton in mass contributions for a of the drop case the in of sensitivity unparticles for running that in element the at loop that we make mass. the As replacement, before, details of the nuclear form factor are encoded in JHEP03(2021)112 (6.5) (6.3) (6.4) (6.1) (6.2) ), this lies below 2.15 , i Λ , ) i p ) ( p ν ( | ν | ) this matrix element z ) ( z p ( N in the insertion approx- ) N . The matrix element in z c ) ( p . z µ ( H . N 4 ) ν H z this matrix element scales as ) M ( m z N N p ( ) must necessarily arise from an p 2 . N 2 λL M M p ν M λL 6.4 2 c m ) )) ν N µ µ y y ( = m ( M 2 c = † c = µ N N 2 N ( 2 c 2 N 2 µ p c ) is added to the inverse seesaw model, there µ 2 EW M 2 µ ) 2 M v N µ 2 – 28 – x ). This is exactly as expected from effective field ) ( M λ 2.19 4 x 2 N p ( 2 EW N 6.2 ) ∼ v M 2 EW N ]. In this framework the rate for neutrinoless double 2 x v ) ( N λ 2 in the insertion approximation. Setting the Higgs to its x ( λ 91 H lies well above 100 MeV, the rate for neutrinoless double , µ ∼ M ) H x ∼ Λ ) 90 ( N . The corresponding matrix element takes the form, x N ( µ f M λL M λL , we see that this contribution to neutrinoless double beta decay T | p ) T | p ) ( p ν ( much greater than 100 MeV, the rate for neutrinoless double beta decay h N ν h N = M = M N N f and M c M µ ) now scales as, ∼ 6.1 We now turn our attention to the opposite limit, If a term of the form shown in eq. ( Before determining the rate for neutrinoless double beta decay in this scenario, it is µ operator of higher dimension than the Weinberg operator. eq. ( For is suppressed compared to thattheory in considerations, eq. ( since the contribution in eq. ( where, as before, we arevacuum treating expectation value (VEV) we see that for from effective field theory considerations. is an additional contribution tonumber neutrinoless violating parameter double beta decay proportional to the lepton scales as Therefore, for in inverse seesaw models is set by the masses of the light neutrinos, exactly as expected where we are treatingimation. the lepton number Setting violating the parameter Higgs to its VEV, we see that for useful to firstbeen consider studied this in, process forbeta in example, decay [ the depends context onpropagator. the of lepton inverse As number seesaw can violatingdepends be models. contribution on seen to a the from This matrix off-shell the element has of neutrino Lagrangian the for form, the inverse seesaw, eq. ( that if the compositeness scale beta decay iseffective determined theory, by just as the ina mass conventional 100 high MeV, matrix the scale of situation seesaw models. isbe the more taken However, complicated, light into if and account neutrinos the when in effects calculating of the the the rate low strong for dynamics energy neutrinoless must double beta decay. dimension term that can be added to the SM which violates lepton number, it follows JHEP03(2021)112 ], (6.8) (6.9) (6.6) (6.7) ) will (6.10) 93 , ) using 6.7 92 , the rate 6.7 Λ . i ) p p ( 2 ν | / ) i c ) 2 ) is also present, the / . We can estimate the z 2N ( Λ ∆ 6.3 N +1) − c ) at scales N O ) 2N } . p z 6.7 ∆ (2∆ ( II 2 − H N + 2 12 ) N 2 I x z p ( / { (2∆ x ). This example serves to illustrate M 2 / , the matrix element in eq. ( − = ˆ λL 2 12 6.5 Λ 3 i c 2 x ) . Then, using the fact that the neutrino ) +1) / µ x / y x µ 8 3 c ( 2 ), it follows that in inverse seesaw models, ν x c / − 2N ( ≤ c 3 N m c 1 c / 2 6.2 x (∆ 2N / x 2N = limit, provides an additional contribution that O ∆ 2N − c 2 23 O – 29 – p µ 1 µ ) x 2 23 / 2 x ) ˆ +∆ x 2 x 2 x 2 EW ( p / ( N v 2 N N 2 / N c λ O +1) m 2∆ O ) c 2N ) 1 ∼ 2N x ∆ x ( ( N (∆ N 2 13 H lies well above 100 MeV, both the neutrino masses and neutri- ) f . However, if the contributions to the neutrino mass are to be x M 2 13 c x hO Λ ( ), in the x 1 , we see that this contribution to neutrinoless double beta decay, 2N C p 2 ˆ λL 6.3 O h C = T I = | ) and N lies below 100 MeV, the situation is more complicated. While the neutrino p II are undetermined coefficients that depend on the details of the CFT and ( N M Λ ν 2 O h C = and . Based on this, we see that the corrections to the neutrino propagator from c U µ and µ ) are either ultraviolet divergent or infrared divergent, depending on the scaling x 1 M . At energies below the compositeness scale µ ∼ C ¯ σ Λ 6.7 If the scale We now consider the scenario in which the singlet neutrinos are composite. Both the µ ≡ / Here x eq. ( dimensions of finite in the ultraviolet, we require and rate for neutrinoless double betathe decay general by expression evaluating for the matrix the element fermion-fermion-scalar in 3-point eq. function ( in a CFT [ where ator, effective field theory dictatesdetermined that by the the rate form for of neutrinoless the double mass beta matrix decay formasses will the still be depend light on neutrinos. thefor value of neutrinoless the double matrix beta element decay in is eq. now ( controlled by scales If the compositeness scale noless double beta decay dependp on the value of thisgenerate matrix element nonrenormalizable evaluated at lepton momenta Since number the violating Weinberg operator interactions is the among unique the lowest dimension SM lepton number fields. violating oper- Majorana masses of thearise from light the neutrinos lepton andcomposite number the violating sector. contributions rate These to for are the controlled neutrinoless neutrino by double propagator the from beta matrix the element, decay For although also suppressed, is largerthat than if that the in scale eq. atrate ( which of the neutrinoless neutrino double masses beta are decay generated need lies not at be or controlled below by the 100 MeV, Weinberg the operator. if the mass ofbeta the decay singlet is neutrinos suppressed.matrix lies element in below If eq. 100 a MeV, ( scales the term as, rate of for the neutrinoless form double shown in eq. ( Comparing this to the matrix element in eq. ( JHEP03(2021)112 be- Λ (6.11) disfavor scenarios eff N less than 100 eV, the partially GeV, the couplings between the 2 Λ . We shall focus on the process / 1 , the rate for neutrinoless double c U . 2N ν ] that can be translated into a lower ∆ Λ − 97 N ∆ U ↔ 100 MeV − ν 4 2 2 , the rate for neutrinoless double beta decay is Λ p and Λ ]. For values of – 30 – 96 ν to estimate the prefactor, we find that the matrix – m below an MeV are disfavored. Λ 94 ∼ Λ U ↔ U 100 MeV U ν . We find that the CMB bounds on ∼ < , M eff Λ N ↔ U ν ν , we see that the contribution to neutrinoless double beta decay from 8 GeV, values of ) scales as ≤ 1 c 6.7 . are suppressed. 2N Λ Λ +∆ N , which does not require an initial population of unparticles. 2∆ Cosmological observations can be used to place constraints on the compositeness scale In the early universe the hidden sector is in thermal contact with the SM, which popu- It follows from this discussion that for → U . Precision measurements of the CMB have established that the neutrinos are free stream- νν scale for any 7.1 Cosmological history In the early universesector. there These are include several processes that can serve to populate the hidden radiation during the epochbound of on acoustic the oscillations compositeness [ scale.fective number These of bounds neutrinos, are usuallyin expressed which as the limits hidden onMeV, sector the the is ef- scale in at equilibrium whichnos with the are the neutrinos in neutrinos decouple equilibrium from at with the temperatures rest the below of hidden an the sector SM. at temperatures Since the of neutri- order the compositeness Λ ing at temperatures of ordercomposite an character eV of [ theto neutrinos admit leads free to streaming. neutrinolow Therefore self-interactions 100 these that eV. observations are disfavor In too compositeness addition, large scales the CMB also places tight limits on the total energy density in decay into final statesabout that 50 may MeV, include we charged findthe leptons. that photons For in these compositeness the decaysin scales SM may distortions below thermal occur to bath the late have CMB enough that gone that may out be they of large happen chemical enough equilibrium. after to be This observed results in future experiments. lates the states in the hiddenSM sector. and We the find hidden that sector for equilibrium are at the large compositeness enough scale. thatcomposite At the temperatures singlet two below neutrinos sectors the annihilate compositeness are scale, awayonly necessarily the efficiently an in into exponentially thermal light suppressed neutrinos, relic leaving population. behind These relic singlet neutrinos eventually 7 Cosmology In this section wethe explore limits the that cosmological cosmology historythis places of scenario on this can the class lead parameter of to space models. observable signals of We in these determine the theories CMB. and show that beta decay is determinedseesaw by models. the However, neutrino for masssuppressed matrix, by just form as factors, in and may conventional be high below scale the reach of next generation experiments. element in eq. ( Since scales above mass is generated at scales of order JHEP03(2021)112 , 2 Λ MeV (7.1) (7.2) , and / 1 2 ), when π ), taking Nν & 16 2.6 Λ → 2.18 O mixing, can take NN in the final state. N ν GeV this process by 1 − . as ν ∼ > 2 Λ T Λ 4 . 5 4 N EW Λ T . This allows us to estimate the of order M 4 ), this is not sufficient to overcome λv ). As emphasized earlier, the large T 7 NN by noting that, at momentum scales to its lower bound in eq. ( 2 N 2.20 λ EW → 2 M κ λv Λ → U νν 2 , eq. ( , but also semi-annihilation π GeV. For values of π νν 1 κ 4 4 N 1 νν – 31 – . Setting , the cross sections for these processes diminish , we see that this process will necessarily bring the ∼ . Λ Λ i ∼ → Λ v Λ . Once the temperature falls below the composite- ∼ at temperatures ∼ Λ NN T →U , the cross section for this process is expected to be of → NN N Λ νν νν legs. The conventional annihilation process begins from M → U σ σ of order h . Since the initial state only involves SM neutrinos, which ν ν 2 ∼ νν T , when n π Pl . Since the mixing between the SM and composite neutrinos is 16 →U /M νν νν ∼ 2 σ , while the dimension-6 self-couplings, which are of order κ T ) → ), and therefore do not leave any observable cosmological effects. N ∼ νN H /M 3.13 EW the composite singlet neutrinos decay prior to neutrino decoupling, as can be eV and recalling that λv Λ ( 01 . O 0 We expect that the hidden sector transitions to the hadronic phase at a temperature We can estimate the inclusive rate for ∼ of order the compositeness scale ν It is therefore mixing-suppressedannihilation and process, subdominant although relative suppressed to byadvantage semi-annihilation. even of more the The powers background co- of and bath mixings of allowed relativistic by existing SMthe constraints neutrinos. (see suppression figure from However, mixing. for conventional annihilation process co-annihilation small, are sizable at temperatures with the number ofthe external same initial state as semi-annihilation, but involves an additional sector is in thesector arise hadronic from phase, the thecombined mixing with dominant between the interactions the self-interactions SM betweenself-interactions of and the the between composite SM the neutrinos and indistinguish singlet hidden eq. it neutrinos ( from are conventional characteristic inverse of seesaw this models. framework These and give rise to not just the T ness scale, the hadronsthe in composite the singlet hidden neutrinosdensity sector occurs is primarily begin relatively through to slow, annihilation go to and SM out so particles. of the When the reduction the bath. strongly in coupled their The number decay of two sectors into equilibrium provided itself may not bescale. sufficient to Nevertheless, bring it thevalues will two of partially sectors in populate equilibrium theseen below CFT from the states. eq. confinement ( However, for these higher This process will ensure thattemperatures the of hidden order sector the is compositenessexpansion in rate, scale equilibrium provided with the the ratem SM is neutrinos faster at than the Hubble Here the parameter are ultrarelativistic, the rate for this process can be estimated as, of order the compositenessroughly scale the same sizecross as section the for cross the section process for JHEP03(2021)112 . ]. Λ 96 – (7.7) (7.6) (7.3) (7.4) (7.5) 94 ) and used the fact , 5 4 2.18 Λ T . 4 3 ν N . ν is of order, N m M Λ m m . Λ 2 . π 2 4 at freeze out is of order κ 4 F F Λ T T 2 π N | /T 8 ∼ > κ 4 N H below 100 GeV. This shows that the M & = 4 N − 2 EW N EW 5 4 e v Λ F m 2 M 2 Λ T , by equating the annihilation rate at freeze T λv / λ F 8 3 i| ) 2 n π v 2 κ F 4 π – 32 – κ 4 T Precision measurements of the CMB have estab- νN N EW ∼ N to its lower bound in eq. ( → m ∼ , for all F λv M λ T ( νν NN 15 i| σ / 2 v → ∼ π h N κ 4 . We can obtain an estimate of the relic density of com- F νν F Nν n σ M n ∼ → Nν . νν . NN → Λ → F σ h T νν σ ν NN n . The non-relativistic number density of Λ ∼ denotes the temperature at freeze out. We estimate the annihilation rate as, N F epoch of acoustic oscillations.compositeness scale These can be translated into a lower bound on the The sizable self-interactions thatcan prevent arise the from neutrinos from the freeby streaming partial CMB during compositeness the measurements, CMB leading of era. to neutrinos This a is lower disfavored The bound CMB on also places bounds on the total energy density in radiation during the T M We therefore focus on the depletion of composite singlet neutrinos through the semi- • • Then the inverse of the mean free path for neutrino scattering is of order, lished that the neutrinos are freeThis streaming during can the be epoch used of to acousticmean oscillations place free [ limits path on of theatures the size of of neutrinos neutrino must order self-interactions. be anscattering In greater eV. at particular, than temperatures the In the below size our the of compositeness framework, the scale the universe cross at section temper- for elastic neutrino-neutrino We now consider these effects in turn. Limits on neutrino free streaming. rise to observable effects. 7.2 Cosmological boundsCosmological on observations the can compositeness be scale There used are to two place effects constraints that on we the need compositeness to scale consider. From this we findnumber that density of relic compositeatures singlet below neutrinos the is compositeness exponentially suppressed scale. at However, temper- as we shall see later, these may still give Here To obtain the inequality wethat have set annihilation process, posite singlet neutrinos, which we denoteout by to the Hubble expansion rate, JHEP03(2021)112 . , . Pl eff N N (7.8) /M M 2 T ∗ g ). To satisfy = ). For H 2.18 2.18 ]. The current upper ), 97 ]. This can be used to , 98 3.13 ν m 4 3 N π M 2 F 192 G MeV. 1 & The predictions of Big Bang Nucleosyn- & 5 N Λ M to its lower bound in eq. ( to its lower bound in eq. ( 2 λ λ ]. However, when the composite states go out – 33 – N EW 101 M – λv 99 3 π 2 F are the neutrinos. When the strongly coupled sector comes G eV. 192 Λ . As we now explain, this constraint disfavors compositeness ∼ 5 100 . N 3 & Γ . Λ eff N less than an MeV, even in the case that the hidden sector comes into equilibrium Λ There are also strong limits on the total energy density in radiation during the epoch of GeV the composite singlet neutrinos decay when the temperature of the SM bath is below where the inequality is1 obtained by setting an MeV. This could potentially affect the successful predictions of BBN by dissociating 7.3 Spectral distortionsThrough of their the mixing CMB with from theally SM late decay neutrinos, decays through the the relic compositedecay weak width singlet interactions neutrinos of into eventu- the final singlet states neutrinos composed can of be SM estimated particles. from eq. The ( SM prediction. The sizehidden of sector. this Unless effect this depends numberdegrees on is of the very freedom), number small this of (less class than degreesWe of the therefore of theories equivalent focus freedom of is in our 2 disfavored the Weyl attention by fermion on the current the CMB regime bound on with at temperatures of order into equilibrium with thecomoving neutrinos, entropy the density comoving increases energy [ of density the bath, is the conserved, comovingincreases. while entropy density the Therefore, is if conserved, the whilein the compositeness the comoving scale energy total lies density comoving below energy an density MeV, in we expect the an light increase neutrinos at the time of CMB over the bound stands at scales with the neutrinos only atthe temperatures rest below of an the MeV.below SM an The bath MeV, SM at then neutrinos temperatures the decouple only of from SM order particles an that MeV. the composite If sector the will compositeness be scale in lies equilibrium sector exit the bath priorpositeness to scale BBN. lies In below principle,temperature an the than MeV, bound the provided can SM that also and be the comes satisfied hidden into if sector equilibrium the is com- withacoustic initially the oscillations at neutrinos based a on only lower precision after measurements BBN. of the CMB [ place a strong limitthe SM on at any temperatures additionalthe of energy rest order density of an MeV, the inwith the SM. radiation the time from SM at This at which physics boundif the temperatures beyond implies the neutrinos of that compositeness decouple order the scale from an hidden lies MeV. above sector This an cannot constraint MeV, be so is in that automatically equilibrium the satisfied hadrons of the strongly coupled the constraint on freeat streaming temperatures this must of be orderscale less an in than eV. this the This scenario, Hubble translates scale, into aLimits lower bound on on the the energythesis compositeness density in (BBN) radiation. in the SM are in excellent agreement with data [ where the inequality is obtained by setting JHEP03(2021)112 . is at Λ ν (7.9) − SN ]. For (7.10) e T + e 104 to represent µ → -distortion of µ N is approximately µ from astrophysical , f Λ ) will hasten the decay . Since N dC τ sec τ 2.18 6 . g 10 , the shift in # f 1 1 × ) ) 7 N N ∼ ∼ < > τ τ MeV, which is many orders of mag- ( ( ≈ x x 1 γ MeV, these effects can be as large as N -distortion. Therefore, in contrast to n n µ & dC 9 50 " τ / 5 Λ x . 2 N − Λ e m 9 GeV / 4 – 34 – 5 / 5 x 2 x 1 / . For . The resulting injection of electromagnetic energy . − 1 1 ν e 2 − ∼ e N f above its lower bound in eq. ( + ≈ τ e 1sec λ ) set to its minimum value, the relic singlet neutrinos decay x → ( 2 λ g 10 N × 8 For an electromagnetic decay fraction ≈ , increasing has been taken as a Lagrangian parameter in this model, here we use 3 µ MeV and N µ ], M 50 103 , . N 102 . Although consistent with current observations, a spectral distortion of this size M 8 − -distortion of the CMB spectrum. 10 For Due to the letter µ 3 ∼ supernova dynamics. In particular,a the emission new of mechanism composite forexplosion singlet the cannot neutrinos core take now place. to offers lose In the energy. remainder If of this thisthe section, energy we loss investigate occurs this question too rapidly, the Since the cosmological bounds alreadynitude constrain higher than the temperature inwill a typical not star, have we expect a thatthe significant neutrino core impact compositeness of on a stellar supernovacomposite is dynamics. of singlet order However, neutrinos 40 the can MeV. temperature Then, be if produced the in compositeness the scale is core, low and enough, could potentially affect the 8 Astrophysics We now turn our attentionconsiderations. to We expect the that limits theastrophysical on effects scenarios the of where compositeness neutrino either compositeness scale are the only temperature important or in the density are comparable to a given value of of the composite singletthe neutrinos other and signals we reduce have the the studied, mixing spectral angle. distortions are more sensitive to lower values of Here the double Comptonone time of scale the is primary givenµ decay by modes, is large enough to be seen at next generation CMB experiments such as PIXIE [ with photon-number conserving processes arenumber still changing processes active at suchAny injection these as of low double electromagnetic energy temperatures, Comptonthe at CMB scattering photon- this spectrum. time are will comparatively manifestgiven itself slow. by in [ a shows that their number density is too small toat have even an later observable effect times,primary on when BBN. decay the channels temperature is into of the the bath bath at is these at late or times below gives a rise keV. to One spectral of distortions the in the CMB. Although nuclei. However, our calculation of the relic abundance of the composite singlet neutrinos JHEP03(2021)112 ) 8.1 (8.4) (8.1) (8.2) (8.3) can be process SN T NN → , ), we find that the νN ) later. SN 2.18 SN /T T N . process. For larger values . ), we find that this process M at a temperature , the inverse of the mean free Λ 3 SN − 4 T 2.18 NN νN Λ 2 N NN , the number density of neutrinos , → , exp ( M ν → 1 1 any composite singlet neutrinos that n 4 → 3 SN λ νN 4 below 800 MeV, the composite singlet T νN , corresponding to a compositeness scale Λ 2 νN N N Λ EW SN SN SN M M L T L λv i 2 i at its upper bound, the corresponding limit on νN 2 π NN to its upper bound in eq. ( – 35 – κ has the effect of increasing the number density 4 N Λ → . The corresponding condition in the case of the λ EW → λ M 3 SN νN λv νN i ≈ σ T σ h NN h ν ν 2 νN π n n κ 4 to its lower bound in eq. ( → → λ νN act to prevent the composite neutrinos from escaping. The i ≈ σ νN h ν 300 MeV. For NN n νN → . → Λ νN σ h ν νN n 10 km is the size of the supernova and and ∼ process takes the form, is large enough. For smaller couplings of 800 MeV. Conversely, if we set λ GeV. SN GeV. Turning our attention to the process . NN 9 νN L 7 at its lower bound, we find that this process prevents the composite singlet neutrinos . Λ 10 1 → The scattering process It follows from this discussion that for all We begin by determining the conditions under which any composite singlet neutrinos , the composite singlet neutrinos may still be prevented from escaping but only if the → λ Λ is . of composite singlet neutrinosis in satisfied, the their supernova. numberseed density Therefore, composite will in singlet increase the neutrinosing exponentially regime are the quickly that initially composite provided present. eq. singlet that neutrinos ( a This into few will thermal and have chemical the equilibrium effect inside of the bring- core Λ neutrinos will be trappedof in the supernova bycoupling the are produced will free stream out of the supernova. For from escaping for all prevents the composite singletscale neutrinos from escapingcomposite provided singlet that neutrinos the remain compositeness Λ trapped in the supernovapath for can all be estimated compositeness as, scales where the exponential suppressionwhere arises the because mass we splitting between are thein in initial the and the supernova final region core. states of is Setting greater parameter than space the temperature νN The inverse of the meanestimated free as, path for the process corresponds to the condition, where inside the core, can be estimated as well above the supernova temperature, and discuss the limit that are produced areνN able to escape freelyrequirement from that the singlet neutrinos supernova. are The trapped scattering in the processes supernova by the further. We will first consider the regime JHEP03(2021)112 , ) λ ), . . λ ) is 8.1 Λ Λ 8.6 (8.6) (8.5) 8.2 . For ) and ( . Once pro- 2.5 GeV. If 8.5 . the dynamics is 4 GeV , they come into ) nor eq. ( 5 GeV Λ λ . 8.1 . Λ . 7 GeV Λ . , the composite singlet . 1 SN SN t t . 3 3 4 GeV Λ ), some number of composite ), some number of composite SN . SN , the number of seed composite L L 2.18 2.18 Λ , neither eq. ( 2 SN 2 λ Nν SN T . T over the lifetime of the supernova can above 1.7 GeV they enter into thermal → Λ SN SN /T νν NN /T . The first process requires the total energy states must occur on the timescales shorter but the amplitude is suppressed by only two N N sec). The number of seed composite singlet M → M N N 800 MeV Nν 2 − – 36 – − 10 ), we find that at least a few composite singlet at its upper bound, the corresponding limit is M e νν e → 2 2 ∼ λ 4 N 2 4 N 2.18 ), we find that at least a few seed composite singlet Λ νν Λ M M SN 6 t 4 2.18 and . For N N EW in the range EW M M NN λv Λ λv is at its lower bound in eq. ( , but the amplitude is now suppressed by an additional power of 4 GeV is at its upper bound in eq. ( → N λ 2 π 2 π λ -number changing process is slow. However, for large enough ∼ < κ 4 κ M 4 νν mixing. For the second process, the total energy in the initial state N Λ N , we expect that their impact on supernova dynamics is limited. If eq. ( − ) ν SN process. In this regime, the composite singlet neutrinos would reach kinetic . /T to its lower bound in eq. ( N λ ) may be satisfied so that the composite singlet neutrinos are trapped by the νN mixing. The production of the seed M at its lower bound in eq. ( 5 GeV 8.2 − → In summary, when There are two processes that seed the initial production of composite singlet neutrinos λ N ∼ < − ing to the limitsinglet when neutrinos are producedduced, inside these the particles supernova core remainthermal provided trapped and inside chemical equilibrium. theequilibrium core. For values but of For not chemical equilibrium. For intermediate values of singlet neutrinos are produced800 MeV, inside these the particles supernovachemical core are equilibrium. provided trapped inside For neutrinos the are able supernova, to stream and freelywe out come find of that into the supernova. the thermal resulting However, from and energy eqs. loss ( is too small to affect the supernova dynamics. Turn- For neutrinos will betimescale produced provided through thisΛ process in the supernova core on the relevant is at its upper3 bound, GeV. the Turning corresponding our uppersinglet attention neutrinos bound produced to on can the the be compositeness process estimated scale as, is be estimated as, Setting neutrinos are produced in the supernova through this process provided powers of the particles need only be ν than the lifetime ofneutrinos a produced through supernova the ( process satisfied, and any composite singletsupernova. neutrinos However, that as are we produced now willsuppressed argue, free in and stream this the out regime resulting of their energy production the loss rate is is exponentially too smallinside to the supernova, affect the supernovain dynamics. the incoming neutrinos to be at least exp ( is not satisfied,eq. the ( νN but not chemical equilibrium. For smaller values of of the supernova. However, because their equilibrium number density is suppressed as JHEP03(2021)112 , there is no SN T Λ . In this regime composite singlet neu- SN T . Λ – 37 – , once the trapped singlet neutrinos approach thermal equilibrium Λ & SN T . The primary effect of the chemical potential is to make the supernova even more SN T We have outlined a specific realization of this scenario in which, in the ultraviolet, Our discussion up till now has not taken into account the effects of the chemical We now turn our attention to the case when the compositeness scale is less than or the composite sector istain a realistic strongly neutrino coupled massesconsequence CFT. in of the The the framework small of scalingbe parameters a dimensions small necessary low-scale of to either seesaw operators ob- because naturally in arise the the as mixing CFT. a with composite Neutrino states masses is could dynamically suppressed, or strongly coupled, the phenomenologyconventional of inverse this seesaw. class For of example,can models now the differs result decay in greatly multiple of from singletinteractions SM that neutrinos of particles rather of the than to a just composite the one. singletthe hidden neutrinos In early sector addition, allow universe, the so them large that to self- below the annihilate severe a away cosmological GeV efficiently bounds are in on naturally singlet satisfied. neutrinos with masses then partially composite particles.sector allows An the explicit neutrinos breakingspective to this of obtain framework lepton leads small number tomasses Majorana in an of masses. inverse the the seesaw hidden model singlet Fromenergies for the neutrinos above neutrino are low the masses set energy compositeness in by per- scale, which the the where compositeness the constituents scale. of However, the if singlet probed neutrinos are at 9 Conclusions We have proposed a framework inof which the a SM strongly neutrinos mix coupled with hidden the composite sector fermions through the neutrino portal. The light neutrinos are neutrinos, increasing their netscales abundance of inside order the 200the MeV supernova supernova or core. dynamics. lower, this Forconstraint. However, compositeness it effect We defer is may a once be carefulfuture study again large work. of not enough the clear effects to that of significantly this this affect class leads of to models on a supernovae robust for potential of the electron neutrinosthen in the supernova core.opaque This to is composite of order singletcontribution 200 to neutrinos, MeV, supernova higher and cooling from so thesome the hidden part conclusion sector remains of that unaltered. the there Now chemical is however, potential no could significant be passed on to the trapped composite singlet with the SM inside theback supernova, the to composite the dynamics conformal may undergosizable, phase. a this phase If could transition the potentially numbergiven have of that a at degrees large present of our impact freedomuse understanding on this in of the to the supernova explosions supernova place hidden is dynamics. a sector limited, robust is However, it constraint. is difficult to significant contribution to supernova cooling from the hidden sector. comparable to the supernova temperature, trinos are easily produced, andhowever, since once produced, will be trapped inside the supernova. Now expected to lie between these two extremes. We conclude that for JHEP03(2021)112 (A.1) being W boson with , ) W W ( µ disfavor this scenario. N eff 2 / N N N ) 5 γ − are the spinors of the charged lepton and conversion at rates that can be probed in (1 e 1 2 N µ → ¯ `γ – 38 – 2 µ and / 3 ¯ ` − EW N and ∆ UV ˆ λv is the polarization vector of the eγ M ) 2 g → √ W ( µ µ = , the matrix element is, U M i ` gauge coupling. → W SU(2) is the g We have shown that this class of models has important implications for a wide range of where the unparticle state, and its 4-momenta. A Matrix elements andFor widths the process and ZL would like toScience thank Foundation grant the PHY-1607611, Aspen where CenterRH part for would like of Physics to which the thank is studywas the supported completed, was Galileo and by performed. Galilei PF National Institute PF would forfor like and hospitality to support. where thank part the PF of Simons andDE-AC02-07CH11359 this Foundation RH (SIMONS, with work 341344 are the AL) supported U.S. by Dept. Fermi of Research Energy. Alliance, LLC under Contract PHY-1914731. ZC is alsoalso supported supported in in part part byto by the thank the US-Israeli the Maryland BSF Fermilab CenterZC’s Grant Theory for stay 2018236. Group at Fundamental Fermilab ZL for Physics. was is supported hospitalityVisiting ZC by during Scholars the would the Fermilab Award like Intensity completion #17-S-02 Frontier Fellowship of and from the this the work. Universities Research Association. PF, RH, be observed in next generation experiments. Acknowledgments ZC and ZL are supported in part by the National Science Foundation under Grant Number violation processes such as upcoming experiments such asthe COMET pion and mass Mu2e. or Ifform below, the factors. the compositeness rate scale For ofsinglet is compositeness neutrinoless neutrinos of scales double can order below beta give rise 50 decay MeV, to will the spectral be late distortions suppressed in decays by the of CMB relic that composite are large enough to experiments, including colliders and beamneutrinoless double dumps, beta searches decay, for and leptonparticles precision observations flavor can of violation now the and decay CMBThese spectral to particles shape. final can SM states be thatand long-lived, include resulting beam multiple in dumps. composite striking displaced singlet vertex neutrinos. At signals loop at colliders level, the composite sector can contribute to lepton flavor ness scale, with a continuumlepton of number possibilities symmetry between could these also twoa arise extremes. as concrete The a approximate but consequence of verynomenology. dynamical broad effects. In range particular, This of gives theBelow possibilities scale an for of MeV, the neutrino we find compositeness compositeness that can scale the be and cosmological as for limits low on phe- as an MeV. because lepton number is an approximate symmetry of the hidden sector at the composite- JHEP03(2021)112 are 2 (A.6) (A.7) (A.8) (A.9) (A.2) (A.3) (A.4) (A.5) / 1 (A.10) − N + 7 ∆ 2 N A . ∆ ` W δ ) E 4 0 2 m δ ( − is cumbersome, O + 1) 1) ) ` ` + ! N ` − , δ E W 5 2 E 2 E N 2 N δ 2 W 2 − 2 m m , N . − m d − (∆ ) 1)( ∆ ` 1)(2∆ f ! ` W δ ! , δ W − E ) ) E − 2 + N 2 W m 2 2 IR δ 2 N 2 W m 1 2 2 N , 3
N µ m m − ν m , the integration over lepton and ) (∆ ∆ − − ` 2(2 . f 2 N 2 , − W 2 W W E N − / W − µ t 1 ` × m m 2 + 5 2 m W W − a 3)(2∆ m 2 W E 3 t
− / N m 2 and the phase volume factor m Γ( Γ(∆ + 2 1 ) 1 + − ∆ ` 2 + ) + 2) W and the matrix element squared becomes, − N δ 2 W N N − E A 3 N ` dt ∆ N − m , δ · µν m 3 1 δ ∆ z − ( E 3 N g ` − − 2 0 2
( 2 W A − ) | (2∆ N Z 3 2 N − W θ 2 3 N (∆ δ – 39 – 2 N m 1)(∆ 2 5 2 π m 2 m − 2∆ f = / 2 | EW 3 2∆ m | 2 N UV 5 − π ) = = 3 ∗ ν − − − 2 a ˆ λv − − M N N 2∆ EW | ( N W µ N EW . The full expression for 2 N 2 N z M ∆ p 2∆ 2 . 2 W m ˆ ∆ UV λv 2 W B ˆ λv | 2∆ g ! UV | ) = ) = (16 W 4(∆ m M M X 3 2 1 pols 2 W 2 m /m = M 2 , δ , δ 2 IR | = − − 2 2 N N µ 2 m δ 1 2 IR N g g 2 W 2 N 2 µ ∆ EW B | (∆ (∆ − ) m M| = = 5 2 m | f f δ ≡ ˆ λv ` 2 W − 1 2 1 2 | 2 EW W δ E N − m − − 2 2 m W ∆ ˆ λv N N π | M| δ | ∆ ∆ m 4(1 − 2 2 W A A 192 π 1 − +2 g 0 3 2 m
1 2 → 0 rest frame → lim are the 4-momenta of the corresponding states, and we have ignored the mass δ 192 g Z × N lim ) = is ∆ N W = , δ ) a N ( Γ = z and (∆ B ` f The only kinematic quantity this matrix element depends on is the lepton energy In evaluation of the spin and polarization averaged matrix element squared, we use . In the ` . Using the rest frame relation ` where Some interesting limits for the product of where, we have defined E unparticle phase space can be carried out, of where the identity, and we obtain, JHEP03(2021)112 (A.15) (A.16) (A.11) (A.12) (A.13) (A.14) Conf. (1977) , , 2 67 ! , Cargèse e , 2 m m 2 IR ` E m m µ m 2 E , m 2 . N d 2 ) / ∆ ! δ 1 (
2 m − 2 IR g O N . m µ Phys. Lett. B ∆ × + , − , O. Sawada and A. A 2 ) / 3 ` 1 2 1 m + 1) − , in the proceedings of the Quarks and leptons E − 2)+1 N m − N 2 ]. N / ]. 1 N N ∆ 2∆ − , in 1 2 − ∆ A N 1 + − 3 2 N
1)(2∆ m SPIRE (∆ ) , − SPIRE UV θ )Γ(∆ ∆ 2 is, muon decays? N 2 Φ IN m 3 2 ) − IN 2 ) π M d [ 9 / δ δ U ) 2∆ ][ 5 N + ` e 10 − − (16 − 2 E N N | Complex Spinors and Unified Theories → m ∆ (2 (1 UV 2 ! π m EW 3)(2∆ m | 16(1 M Γ(∆ – 40 – 2 m 2 IR − ˆ λv (1980) 912 | EW m µ Neutrino mass and spontaneous parity 2 2 N m | 3 ) = 2 ) = 2 f ˆ 44 λv − | − 2 , our calculations agree with the standard results for π limit. | , δ , δ 2 ` N m 0 EW (2∆ m N N N f 2 32 E qq 2 2∆ / ˆ m λv UV arXiv:1306.4669 2 | M V | ), which permits any use, distribution and reproduction in 0 3 | (∆ (∆ in the unparticle treatment is identified with the mass of 2 m ) = M g g qq − 2 F 2 f 1 2 1 2 → V 2 , δ 1 π G | | IR − − 0 N m at a rate of one out of
N 2 F µ N N 32 qq G m ∆ ∆ ∆ × (∆ V eγ | A A g 2 F = Phys. Rev. Lett. 0 (1979) 315 [ 3 2 → CC-BY 4.0 , G ) = → → µ lim The future of elementary particle physics δ m U Horizontal symmetry and masses of neutrinos N lim This article is distributed under the terms of the Creative Commons ]. ` m ∆ → 790927 ) = m SPIRE U is the mass of the charged meson. IN e Γ( [ d m → m Proc. C lectures, M. Lévy et al. eds., Plenum Press, New York,nonconservation U.S.A. (1980). 421 Workshop on unified theory andSugamoto baryon eds., number KEK, in Tsukuba, Japan the (1979). universe m S.L. Glashow, R.N. Mohapatra and G. Senjanović, P. Minkowski, T. Yanagida, M. Gell-Mann, P. Ramond and R. Slansky, After integration, the partial width of For meson decays, neglecting the lepton mass, the partial width is given by [4] [5] [1] [2] [3] Γ( Attribution License ( any medium, provided the original author(s) and source areReferences credited. Open Access. with In this case the two limits of interest are, where the composite singlet neutrinos elementary HNLs in the When the infrared cutoff JHEP03(2021)112 , , 22 09 94 , models 08 SO(10) JHEP , U(1) JHEP × (2008) 016 , ]. Phys. Rev. D (1998) 1171 Phys. Lett. B , 12 (2011) 132 , ]. SU(2) 81 01 ]. SPIRE ]. IN ]. ]. JHEP theories SPIRE , ][ IN SPIRE JHEP ][ IN , SPIRE SPIRE U(1) SPIRE ][ IN ]. IN IN × ]. [ [ [ Phys. Rev. Lett. ]. , SPIRE SPIRE SU(2) IN Are there real goldstone bosons associated IN [ ][ SPIRE ]. (1981) 165 arXiv:1105.0291 (1981) 265 ]. Unconventional model of neutrino masses [ (1980) 2860 IN ]. [ 23 arXiv:0901.0006 98 Seesaw neutrino masses induced by a triplet of [ 22 Proton lifetime and fermion masses in an arXiv:1404.7118 SPIRE ]. ]. ]. – 41 – [ SPIRE ]. IN Light active and sterile neutrinos from compositeness SPIRE IN (1986) 1642 [ Neutrino masses and mixings in gauge models with [ Neutrino mass and baryon number nonconservation in IN (2011) 228 Nonconservation of total lepton number with scalar [ SPIRE Composite Dirac neutrinos 34 Left-handed neutrino mass scale and spontaneously broken SPIRE Conformal neutrinos: an alternative to the see-saw SPIRE (1981) 411 SPIRE IN IN IN Neutrino masses in (2009) 317 hep-ph/9806223 IN [ [ 702 ][ 99 [ Phys. Rev. D Phys. Lett. B ]. ]. ][ Neutrino mass problem and gauge hierarchy KeV warm dark matter and composite neutrinos Dynamical framework for KeV Dirac neutrino warm dark , Phys. Rev. D ]. , Leptogenesis with composite neutrinos (1981) 287 678 , Neutrino masses, mixings and oscillations in (2014) 045030 (1977) 433 Constraints on composite Dirac neutrinos from observations of (1989) 441 SPIRE SPIRE 90 70 181 SPIRE ]. IN IN 44 (1986) 561 Phys. Rev. D (1981) 297 IN ]. (1999) 179 ][ ][ , Mechanism for understanding small neutrino mass in superstring theories ][ 56 Phys. Lett. B Phys. Lett. B SPIRE 193 , 459 , IN SPIRE Phys. Lett. B hep-ph/0405083 arXiv:1205.0569 [ , IN [ [ [ Searching for composite neutrinos in the cosmic microwave background Z. Phys. C Phys. Rev. D Phys. Lett. B Pathways to naturally small neutrino masses Nucl. Phys. B , , , , arXiv:0811.0871 arXiv:1009.2781 hep-ph/9805219 (2005) 017 Phys. Rev. Lett. superstring models galaxy clusters [ [ (2012) 161 matter Phys. Lett. B mechanism lepton number with broken lepton number? Nucl. Phys. B leptons [ of electroweak interactions model spontaneous parity violation bosons (1980) 61 (1980) 2227 R.N. Mohapatra, R.N. Mohapatra and J.W.F. Valle, R.S. Hundi and S. Roy, Y. Grossman and Y. Tsai, T. Okui, D.J. Robinson and Y. Tsai, D.J. Robinson and Y. Tsai, N. Arkani-Hamed and Y. Grossman, G. von Gersdorff and M. Quirós, Y. Grossman and D.J. Robinson, G.B. Gelmini and M. Roncadelli, Y. Chikashige, R.N. Mohapatra and R.D. Peccei, H.M. Georgi, S.L. Glashow and S. Nussinov, R. Foot, H. Lew, X.G. He and G.C. Joshi, E. Ma, G. Lazarides, Q. Shafi and C. Wetterich, R.N. Mohapatra and G. Senjanović, T.P. Cheng and L.-F. Li, M. Magg and C. Wetterich, J. Schechter and J.W.F. Valle, W. Konetschny and W. Kummer, [9] [7] [8] [6] [25] [26] [22] [23] [24] [20] [21] [17] [18] [19] [14] [15] [16] [12] [13] [10] [11] JHEP03(2021)112 , 830 08 Int. J. Phys. (2004) , Phys. , ] , Phys. Phys. Nucl. Phys. Lett. , , , 583 , JHEP (2008) 070 , (2014) 2766 (1998) 231] 01 (1998) 253 2 , Germany (2016) (2007) 275 74 2 Erratum ibid. [ ]. 650 JHEP , hep-ph/0703260 Phys. Lett. B SPIRE Springer (2004) 058 [ standard model , , IN D (2007) 157 ][ 11 − 5 ]. 768 Eur. Phys. J. C ]. ]. ]. , Phys. Lett. B JHEP ]. , , ]. SPIRE ]. (2007) 221601 SPIRE SPIRE IN SPIRE Adv. Theor. Math. Phys. Gauge theory correlators from noncritical Colored unparticles IN Adv. Theor. Math. Phys. IN ]. IN SPIRE ][ 98 Holography and phenomenology ][ , SPIRE ][ ]. ][ IN ][ SPIRE hep-th/9802109 IN ][ [ IN ][ Nucl. Phys. B SPIRE ][ , SPIRE IN ]. – 42 – ]. [ IN Composite Higgses ][ Warped seesaw mechanism is physically inverted SPIRE (1998) 105 SPIRE ]. IN Neutrino masses and mixings in nonfactorizable geometry hep-ph/9912408 IN Phys. Rev. Lett. AdS/CFT correspondence and symmetry breaking limit of superconformal field theories and supergravity hep-th/9711200 ]. Comments on the holographic picture of the Randall-Sundrum A large mass hierarchy from a small extra dimension [ hep-th/0012248 ][ , hep-ph/0312392 Holography for fermions ]. ]. ]. ][ [ 428 [ [ (1991) 259 ]. ]. N hep-ph/9905221 arXiv:1512.06742 The composite nambu-Goldstone higgs [ SPIRE Seesaw mechanism in warped geometry Majorana neutrinos in a warped [ arXiv:1010.2659 hep-th/9905104 SPIRE IN 365 [ [ SPIRE SPIRE SPIRE IN ][ SPIRE SPIRE IN IN IN ][ IN IN hep-ph/0205327 (2000) 361 ][ ][ ][ [ (2001) 021 ][ ][ The large (1999) 1113 Light neutrinos from a mini-seesaw mechanism in warped space Dirac neutrino masses with Planck scale lepton number violation 474 04 Flavor at SSC energies: a new mechanism for dynamically generated fermion Phys. Lett. B (1999) 3370 (2004) 161601 38 (1999) 89 Neutrinos in a sterile throat , (2011) 266 hep-ph/0611218 hep-th/0012148 Anti-de Sitter space and holography (2016) 013001 Unparticle physics Another odd thing about unparticle physics [ 83 92 ]. ]. 556 Nucl. Phys. B 696 94 (2002) 295 JHEP , , hep-ph/0309252 [ 544 SPIRE SPIRE arXiv:0708.0005 hep-th/9802150 IN hep-th/0406257 IN arXiv:0704.2457 arXiv:1401.2457 arXiv:1506.01961 Lett. B (2010) 390] [ [ 293 Rev. Lett. Rev. D Phys. Lett. B B (2001) 017 model Rev. Lett. [ Phys. B Theor. Phys. [ string theory [ [ [ [ [ masses B. Gripaios, G. Cacciapaglia, G. Marandella and J. Terning, T. Gherghetta, K. Agashe, S. Hong and L. Vecchi, K.L. McDonald, Y. Grossman and M. Neubert, S.J. Huber and Q. Shafi, S.J. Huber and Q. Shafi, R. Rattazzi and A. Zaffaroni, L. Randall and R. Sundrum, E. Witten, I.R. Klebanov and E. Witten, N. Arkani-Hamed, M. Porrati and L. Randall, J.M. Maldacena, S.S. Gubser, I.R. Klebanov and A.M. Polyakov, R. Contino and A. Pomarol, H. Georgi, H. Georgi, G. Panico and A. Wulzer, D.B. Kaplan, B. Bellazzini, C. Csáki and J. Serra, [46] [47] [43] [44] [45] [40] [41] [42] [38] [39] [35] [36] [37] [33] [34] [30] [31] [32] [28] [29] [27] JHEP03(2021)112 JHEP JHEP Nucl. ]. , , , ]. Z. Phys. C Nucl. Phys. ]. NTN Phys. Lett. B SPIRE , , , IN SPIRE ]. ][ IN ]. SPIRE ]. bosons produced in ][ IN [ , W SPIRE (2019) 095022 SPIRE (2015) 059 IN SPIRE IN ][ Phenomenology of IN 05 (2018) 221801 ][ 100 ][ ]. . ]. (1979) 327 120 JHEP Sensitivity of lepton number SPIRE 96 arXiv:1805.08567 , [ SPIRE IN arXiv:1708.05389 ]. IN ][ , in the proceedings of the [ , may 29–31, St. Louis, U.S.A. (2019) ][ Phys. Rev. D , Phenomenology of hidden valleys at hadron SPIRE arXiv:1806.05191 ]. Seeking for sterile neutrinos with displaced Physica A IN [ Searching for confining hidden valleys at LHCb, arXiv:1901.09966 ]. ]. (2018) 032 , The search for heavy Majorana neutrinos arXiv:0809.5092 arXiv:1904.01020 [ Phys. Rev. Lett. , Emerging jets [ , 11 – 43 – SPIRE (2018) 095033 SPIRE SPIRE Displaced vertices as probes of sterile neutrino mixing IN TeV IN IN ][ 97 ]. ][ ][ JHEP arXiv:1903.06100 Echoes of a hidden valley at hadron colliders arXiv:0712.2041 (1997) 580] [ , [ (2019) 159 = 13 [ (2018) 035012 ]. ]. ]. s (2020) 010501 75 Chiral quarks and the nonrelativistic quark model 07 ]. SPIRE √ Flavor conserving CP-violation in invisible axion models 98 IN Heavy Neutrinos in displaced vertex searches at the LHC and 47 Search for neutral heavy leptons produced in Z decays [ SPIRE SPIRE SPIRE Search for heavy neutral leptons in decays of (2016). IN IN IN SPIRE Search for displaced leptons in the e-mu channel Search for heavy neutral leptons in events with three charged leptons in JHEP (2020) 070 (2008) 008 Phys. Rev. D ][ ][ ][ IN , Physics beyond colliders at CERN: beyond the standard model working , [ 02 hep-ph/0604261 07 arXiv:0901.3589 arXiv:1905.09787 [ Erratum ibid. [ [ J. Phys. G (1986) 241 Phenomenological lagrangians [ Phys. Rev. D , A review of unparticle physics , collisions using prompt and displaced signatures with the ATLAS detector collaboration, JHEP JHEP collaboration, ]. 276 , , pp (1984) 189 collaboration, collaboration, (2007) 374 (2019) 265 (1997) 57 (2009) 030 234 SPIRE arXiv:1502.05409 IN arXiv:1908.09562 arXiv:1802.02965 colliders [ ATLAS, and CMS GeV-scale heavy neutral leptons 651 at the LHC Neutrino non-standard interactions: aWorkshop status on report Neutrino Non-Standard Interactions [ HL-LHC violating meson decays in[ different experiments group report leptons at the LHC CMS CMS-PAS-EXO-16-022 ATLAS 13 TeV 10 DELPHI 74 05 CMS proton-proton collisions at [ B Phys. B P. Schwaller, D. Stolarski and A. Weiler, A. Pierce, B. Shakya, Y. Tsai and Y. Zhao, K. Bondarenko, A. Boyarsky, D. Gorbunov and O. Ruchayskiy, M.J. Strassler and K.M. Zurek, T. Han, Z. Si, K.M. Zurek and M.J. Strassler, G. Cottin, J.C. Helo and M. Hirsch, E.J. Chun, A. Das, S. Mandal, M. Mitra and N. Sinha, J. Beacham et al., J. Liu, Z. Liu, L.-T. Wang and X.-P. Wang, M. Drewes and J. Hajer, A. Atre, T. Han, S. Pascoli and B. Zhang, A. Manohar and H. Georgi, H. Georgi and L. Randall, M. Kenzie, S. Weinberg, [66] [67] [63] [64] [65] [61] [62] [59] [60] [56] [57] [58] [54] [55] [52] [53] [49] [50] [51] [48] JHEP03(2021)112 , ]. , , GeV , 5 ] SPIRE Phys. . 0 , IN Phys. Rev. [ ]. , ]. (2020) eν ]. with the full (2018) 00022 γ → SPIRE SPIRE 2020 + π ]. e IN 166 decays to positrons IN [ SPIRE neutrino decay (1985) 105 ][ + → IN + arXiv:1311.2198 e K ][ + 31 , PTEP SPIRE µ arXiv:1605.05081 , → IN arXiv:1501.06858 Proposal: search for decays ]. [ (2018) 1 , + ]. ][ π ]. 731 (1983). SPIRE ]. SPIRE IN EPJ Web Conf. SPIRE ]. IN , ][ (2016) 434 IN Phys. Rev. D ][ Muon conversion to electron in nuclei in arXiv:1501.05241 , SPIRE ]. ][ ]. arXiv:1504.04855 , 76 collaboration, IN [ arXiv:1903.04497 SPIRE ][ [ Phys. Rept. IN [ , A call for new physics: the muon anomalous SPIRE SPIRE arXiv:1209.2679 Second order corrections to the muon anomalous [ IN IN Bounds on unparticles from the Higgs sector Snowmass 2013: Workshop on Energy Frontier [ ][ technical design report – 44 – ]. ]. 2) CERN-PSCC-83-127 , − (2016) 124201 theory value: present and future arXiv:2005.09575 arXiv:1307.7335 Producing and detecting long-lived particles at different g [ arXiv:0705.3092 Eur. Phys. J. C ( (2020) 090501 (2013) 118 SPIRE 79 2) [ , SPIRE (1986) 473 ]. ]. IN IN − 47 01 theory: The hadronic part [ ][ arXiv:1902.07598 g , talk given at ( [ Muon 2 166 arXiv:2004.08820 A search for decays of heavy neutrinos in the mass range COMET Phase-I technical design report , − SPIRE SPIRE Improved search for heavy neutrinos in the decay g The long-baseline neutrino experiment: exploring fundamental JHEP IN IN Search for heavy neutral lepton production in Mu2e technical design report arXiv:1712.03275 Search for the lepton flavour violating decay , Search for heavy neutrinos with the T2K near detector ND280 [ ][ ][ Improved search for massive neutrinos in (2020) 135599 Searching for long-lived particles beyond the Standard Model at the Large A facility to search for hidden particles at the CERN SPS: the SHiP J. Phys. G (2007) 075004 (1992) 885 , The muon Muon 807 76 46 Rept. Prog. Phys. (2019) 052006 , Phys. Lett. B collaboration, , collaboration, collaboration, ]. ]. ]. collaboration, arXiv:1812.09018 100 collaboration, [ collaboration, collaboration, collaboration, GeV (2018) 072012 collaboration, 8 . 2 97 SPIRE SPIRE SPIRE IN arXiv:1705.00263 IN IN arXiv:1610.06587 MEG dataset of the MEG[ experiment type-I seesaw models magnetic moment in alternative electroweak models Phys. Rev. D [ [ COMET 033C01 Mu2e [ magnetic moment and lepton[ flavor violation Muon g-2 physics case LBNE symmetries of the universe July 20–August 6, U.S.A. (2013) [ to NA62 Phys. Lett. B T2K Rev. D PIENU D Annecy-CERN-London(IC)-Paris(LPNHE) of heavy neutrinos with the PSCHARM beam Hadron Collider experiments at the LHC Phys. Rev. D R. Alonso, M. Dhen, M.B. Gavela and T. Hambye, S.R. Moore, K. Whisnant and B.-L. Young, P.J. Fox, A. Rajaraman and Y. Shirman, F. Jegerlehner, T. Blum et al., M. Lindner, M. Platscher and F.S. Queiroz, S. Alekhin et al., C. Yuan, H. Zhang and Y. Zhao, D.I. Britton et al., J. Alimena et al., [86] [87] [83] [84] [85] [81] [82] [78] [79] [80] [76] [77] [74] [75] [71] [72] [73] [69] [70] [68] JHEP03(2021)112 ]. , Rev. , ] , ]. SPIRE (2018) IN ]. ][ 120 SPIRE IN , Springer, ]. ][ ]. Astron. SPIRE , IN (2005) 111801 ][ SPIRE SPIRE IN arXiv:1105.2044 94 IN [ ][ ][ (2019) 1 Phys. Rev. Lett. arXiv:1912.03058 , 7 ]. [ ]. ]. arXiv:1109.6552 ]. ]. (2011) 025 [ Big Bang nucleosynthesis: 2015 SPIRE SPIRE SPIRE 07 arXiv:1209.5342 Late time neutrino masses, the LSND IN IN IN [ Phys. Rev. Lett. Neutrinoless double beta decay versus SPIRE ][ SPIRE ][ ][ , (2020) 170 Conformal invariance in quantum field IN arXiv:1311.3873 IN [ CMB signals of neutrino mass generation [ ][ 03 arXiv:2003.08387 JCAP Front. in Phys. [ (2012) 1294 , Can heavy neutrinos dominate neutrinoless , Constraints on secret neutrino interactions after ]. 419 – 45 – JHEP , ]. (2013) 093007 Updated constraints on non-standard neutrino (2014) 046 Interacting radiation after Planck and its implications (1993) 2661 SPIRE (2020) 029 87 07 hep-ph/0312267 IN 70 [ SPIRE [ arXiv:1505.01076 09 arXiv:1504.04999 The evolution of CMB spectral distortions in the early ]. A search for muon to electron conversion in muonic gold IN [ [ Conformal quantum field theory in D-dimensions ]. ][ arXiv:1807.06209 JCAP [ Thermal dark matter below an MeV , SPIRE Planck 2018 results. VI. Cosmological parameters JCAP , IN The Mu2e Experiment SPIRE (2006) 337 ][ Phys. Rev. D Thermalization constraints and spectral distortions for massive unstable IN (2015) 014 , (2004) 085008 ][ 47 The Primordial Inflation Explorer (PIXIE): a nulling polarimeter for (2020) A6 collaboration, (2016) 015004 07 Phys. Rev. Lett. 70 , 88 641 Mon. Not. Roy. Astron. Soc. collaboration, ]. , arXiv:1706.07046 JCAP collaboration, [ , , Edizioni della Normale, Italy (1978). SPIRE IN hep-ph/0405067 arXiv:1901.11099 Universe Cosmic Microwave Background observations [ 021801 relic particles Phys. Rev. D experiment and the cosmic[ microwave background for the Hubble tension Planck Astrophys. Mod. Phys. interactions from Planck Planck other probes of heavy sterile neutrinos theory Germany (1996). Eur. Phys. J. C Mu2e [ double beta decay? SINDRUM II A. Kogut et al., A. Berlin and N. Blinov, W. Hu and J. Silk, J. Chluba and R.A. Sunyaev, Z. Chacko, L.J. Hall, S.J. Oliver and M. Perelstein, Z. Chacko, L.J. Hall, T. Okui and S.J. Oliver, N. Blinov and G. Marques-Tavares, R.H. Cyburt, B.D. Fields, K.A. Olive and T.-H. Yeh, M. Archidiacono and S. Hannestad, F. Forastieri, M. Lattanzi and P. Natoli, P.D. Bolton, F.F. Deppisch and P.S. Bhupal Dev, I.T. Todorov, M.C. Mintchev and V.B. Petkova, E. Fradkin and M. Palchik, J. Lopez-Pavon, S. Pascoli and C.-f. Wong, [99] [96] [97] [98] [94] [95] [91] [92] [93] [89] [90] [88] [104] [101] [102] [103] [100]