Distinguishing Dirac and Majorana Heavy Neutrinos at Lepton Colliders

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Distinguishing Dirac and Majorana Heavy Neutrinos

at Lepton Colliders

Marco Drewes

1Centre for Cosmology, Particle Physics and Phenomenology,

Universit´e catholique de Louvain, Louvain-la-Neuve B-1348, Belgium

Abstract

We discuss the potential to observe lepton number violation (LNV) in displaced

vertex searches for heavy neutral leptons (HNLs) at future lepton colliders. Even

though a direct detection of LNV is impossible for the dominant production chan-

nel because lepton number is carried away by an unobservable neutrino, there are

several signatures of LNV that can be searched for. They include the angular distri-

bution and spectrum of decay products as well as the HNL lifetime. We comment

on the perspectives to observe LNV in realistic neutrino mass models and argue that

the dichotomy of Dirac vs Majorana HNLs is in general not suﬃcient to eﬀectively

capture their phenomenology, but these extreme cases nevertheless represent well-

deﬁned benchmarks for experimental searches. Finally, we present accurate analytic

estimates for the number of events and sensitivity regions during the Z-pole run for

both Majorana and Dirac HNLs.

Contents

1 Motivation 2

2 Observables sensitive to LNV 3

3 Probing realistic neutrino mass models and leptogenesis 7

4 Practical feasibility and number of events 10

arXiv:2210.17110v2 [hep-ph] 16 Dec 2022

1 Motivation

Neutrinos are the sole fermions in the Standard Model of particle physics (SM) that could

be their own antiparticles, in which case the would be the only known elementary Majorana

fermions, and their masses would break the global U(1)B−Lsymmetry of the SM. An

immediate consequence would be the existence of processes that violate the total lepton

number L. However, due to the smallness of the light neutrino masses mithe rate for

lepton number violating (LNV) processes in neutrino experiments would be parametrically

suppressed.1At the same time it is clear that any explanation of the light neutrino masses

requires an extension of the SM ﬁeld content, and LNV may occur at an observable rate

in processes involving new particles. This in particular can include heavy neutral leptons

(HNLs)2Nithat couple to the Z- and W-bosons and the Higgs bosons hvia the SM weak

interaction with an amplitude suppressed by the mixing angles θαi (with α=e, µ, τ and

i= 1 . . . n),3

L⊃−mW

vNiθ∗

αiγµeLαW+

µ−mZ

√2vNiθ∗

αiγµνLαZµ−Mi

v√2θαihνLαNi+ h.c.,(1)

with mZ,mWthe weak gauge boson masses and v'174 GeV the Higgs ﬁeld vacuum

expectation value. The Nican be Dirac or Majorana fermions. For Mi< mZthey can be

produced copiously during the Z-pole run of future lepton colliders [19] such as FCC-ee

[20] or CEPC [21],4cf. Fig. 1a, making it possible to not only discover them but also study

their properties in suﬃcient detail to probe their role in neutrino mass generation and

leptogenesis [25]. An important question in this context is whether the LNV in Ni-decays

can be observed. This is hampered by two main obstacles, both of which can be overcome,

I) LNV can be detected most directly when the ﬁnal state of a process can be fully

reconstructed, such as W±→`±

αN→`±

α`±

αW∓

∗. However, at lepton colliders Ni

with Mi< mZare dominantly produced in the decays of Z-bosons along with an

unobservable neutrino or antineutrino, making it impossible to reconstruct the ﬁnal

state and determine its total L.

1Neutrinoless double β-decay can provide an indirect probe [1], cf. also [2, 3].

2Here we use the following nomenclature: HNLs are fermions with mass Mmithat carry no charge

under both the electromagnetic and strong interactions. Heavy neutrinos are a type of HNL that mix

with the SM neutrinos. Right-handed neutrinos νRare ﬁelds with right-handed chirality that couple to

the left-handed SM neutrinos with Yukawa couplings and are singlet (sterile) with respect to the SM

gauge groups. They are in general not identical to the mass eigenstates N, cf. footnote 5. In addition to

possible connections to neutrino masses, νRcan potentially play an important role in other areas of particle

physics and cosmology [4, 5], such as leptogenesis [6] as an explanation for the observed matter-antimatter

asymmetry of the observable universe [7] (including low scale scenarios [8–11] that can be tested [12, 13]),

or as Dark Matter candidates [14, 15].

3In general the Nimay have new gauge interactions in addition to the SM weak interactions in (1) that

can also lead to LNV processes (e.g. [16]), but the LHC bounds on the mass of new gauge bosons [17, 18]

make it diﬃcult to explore this option at lepton colliders.

4Linear colliders typically have less sensitivity for M < mZ[22] due to their smaller integrated lumi-

nosity compared to the proposed Z-pole runs at FCC-ee or CEPC [20, 23], but their polarised beams may

oﬀer an advantage when studying forward-backward asymmetries [24], cf. method 1) below.

II) In models that employ the type-I seesaw mechanism [26–31], the light neutrino masses

parametrically scale as5mi∼θ2Mi, while the HNL production cross section scales

as σN∼θ2, cf. (2), so that one may expect σNto be parametrically suppressed by

∼mi/Mi.6This is not the case if the miare protected by an approximate global

U(1)B−¯

Lsymmetry, with ¯

La generalised lepton-number under which the HNLs are

charged [33]. The symmetry would lead to systematic cancellations in the neutrino

mass matrix that keep the mismall while allowing for (almost) arbitrarily large

αi =|θαi|2.

The approximate ¯

L-conservation would, however, also suppress all LNV processes

parametrically. One may expect that the ratio of L-violating to L-conserving Ni-

decays scales as Rll ∼U−2

imi/Miwith U2

i=PαU2

αi and is practically unobservable

even if the Niare fundamentally Majorana particles.

2 Observables sensitive to LNV

Collider studies are often performed in a phenomenological type I seesaw model, deﬁned

by (1) with only one HNL species (n= 1) of mass M. This is not a realistic model of

neutrino mass, but it can eﬀectively capture many phenomenological aspects with only ﬁve

parameters (M, θe, θµ, θτ, Rll),7where Rll = 0 for Dirac-Nand Rll = 1 for Majorana-N.

If all HNLs decay inside the detector the total number of events with n= 1 is the same

for the Dirac and Majorana cases,8but there are at least three ways in which Dirac and

5The type-I seesaw requires the addition of at least nﬂavours of right-handed neutrinos νRwith a

Majorana mass matrix MMto the SM in order to generate nnon-zero light neutrino masses mi. The mass

eigenstates are represented by Majorana spinors νi'[U†

ν(νL−θνc

R)]i+c.c. and Ni'[U†

N(νR+θTνc

L)]+c.c.

with masses miand Mi, respectively. The m2

iand M2

iat tree level are given by the eigenvalues of mνm†

and MNM†

N, with MN=MM+1

2(θ†θMM+MT

MθTθ∗) and mν=−θMMθT.Uνand UNdiagonalise

mνm†

νand MNM†

N, respectively. Strictly speaking θin (1) should be replaced by Θ = θU∗

N, we neglect

this diﬀerence for notational simplicity.

6The precise value of this so-called seesaw line in the mass-mixing plane depends on nand the lightest

mi[32]. If all eigenvalues of MMhave a similar magnitude M, one can roughly estimate the minimal mixing

to be 'ζ∆matm/Mi, with ζ= 1(2) for normal (inverted) ordering of the miand ∆m2

atm '2.5×10−3eV2.

7Practically it is often more convenient to consider the parameters M,U2=PαU2

αand the three

ratios u2

α=U2

α/U2, with α=e, µ, τ . This also gives ﬁve parameters as Pαu2

α= 1. Note that the θαi

for n > 1 are in principle complex while the U2

αi and u2

αi are real (and hence contain less information).

However, the phases only play a role when there are interferences between the contributions from diﬀerent

Ni, which only occurs for ∆M≡ |Mi−Mj| ∼ ΓN, cf. footnote 20.

8Naively one may expect that the number of produced particles is twice as large for Dirac HNLs

(compared to Majorana HNLs), reﬂecting the fact that Dirac fermions have twice as many internal degrees

of freedom. However, only half of them are produced in the decay of a given Z-boson (as Nis necessarily

produced along with ¯νand ¯

Nalong with ν), and one can distinguish two possible types of ﬁnal states that

can be labeled by the light neutrino helicity. The same is true for Majorana HNLs, hence cprod = 1 in both

cases. These conclusions are more general than the speciﬁc process considered here, cf. e.g. [34–37]. The

HNL decay rate ΓN, on the other hand, is twice as large for Majorana HNLs with Rll = 1, as for Dirac

HNLs (Rll = 0) the LNV processes are forbidden. Hence, there are more possible ﬁnal states for Majorana

HNLs which are, however, indistinguishable when simply counting particles because the (anti)neutrino is

not observed. Since all HNLs eventually decay, the total number of events is equal in both cases.

Majorana HNLs can be distinguished at FCC-ee.

1) In the Dirac case a N(¯

N) is always produced along with a ¯ν(ν). The chiral nature

of the weak interaction and angular momentum conservation imply that νand ¯ν

are emitted with diﬀerent angular distributions for a given Z-polarisation. Due to

the parity-violation of the weak SM interaction the Z-bosons at lepton colliders are

polarised at the level of PZ'15%9even if the e±beams are not, hence the angular

distributions of the Nand ¯

Nare diﬀerent [38].10 Since Dirac N(¯

N) can only decay

into leptons (antileptons), this introduces diﬀerences in the angular distribution of

leptons and antileptons. This can be observed in the form of a forward-backward

asymmetry AD

F B 'PZ3

4/(1 −(M/mZ)2/2) ∼10%, cf. Fig. 2a. For Majorana HNLs

there is no forward-backward asymmetry because they can decay into leptons and

antileptons.

2) For the Dirac case, the Nand ¯

Nindividually are highly polarised, cf. Fig. 2b, be-

cause N(¯

N) can only have been produced along with ¯ν(ν), whose helicity is ﬁxed

in the massless limit. Since Ncan only decay into leptonic ﬁnal states ( ¯

Ninto an-

tileptonic ones), the parent particle of leptons and antileptons tend to have opposite

polarisation. The decay rates are polarisation-dependent [37], leading to diﬀerent

spectra for leptons and antileptons [38]. For Majorana HNL there is no diﬀerence

between Nand ¯

N; their polarisaion is of order (and proportional to) PZ, and they

can decay into either leptons and antileptons. This diﬀerence in the lepton spectra

is observable.

3) For long-lived HNLs counting the number of events as a function of displacement

provides an additional probe that is independent of PZ. While the number NHNLα

of HNLs produced in Z-decays along with a lepton or antilepton of ﬂavour αis the

same for Dirac and Majorana HNLs, their decay rate diﬀers by a factor two, leading

to a twice larger decay length in the detector λN=βγ/ΓN, with βγ = pN/M and

pNthe HNL three-momentum. Hence, the number of HNL decays into lepton ﬂavour

βwith a displacement between l0and l1is sensitive to this diﬀerence. It is given by11

Nobs 'u2

βNHNLαexp(−l0/λN)−exp(−l1/λN)αβ,(2)

9PZ= (g2

L−g2

R)/(g2

R+g2

L)'15% with gL= (1−2 sin2θW) and gR= 2 sin2θWthe left- and right-chiral

neutral current charges of the charged leptons, respectively, and θWis the Weinberg angle [38].

10 For Dirac HNLs one ﬁnds diﬀerential production cross sections for e+e−→Z→N¯νand e+e−→

Z→¯

Nν [38]

σN, ¯

dσN, ¯

dcθ

4(g2

R+g2

(2m2

Z+M2) g2

R(1 ∓cθ)2+g2

L(1 ±cθ)2+M2

(g2

R+g2

L)s2

θ!,

with with cθand sθthe sine and cosine of the angle between the HNL and electron momenta. For Majorana

HNLs the angular distribution is given by the sum of the diﬀerential Nand ¯

Nproduction cross sections.

11The simple analytic estimate (2) can even describe the number of events in proton collisions surprisingly

well if it is weighted by an appropriate momentum distribution that has to be obtained from simulations

[39, 40]. For the Z-pole run at lepton colliders (2) is even more accurate, and the sensitivity region can

be described analytically by (4), (3a), (3b), cf. Fig. 1b.

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DistinguishingDiracandMajoranaHeavyNeutrinosatLeptonCollidersMarcoDrewes1CentreforCosmology,ParticlePhysicsandPhenomenology,UniversitecatholiquedeLouvain,Louvain-la-NeuveB-1348,BelgiumAbstractWediscussthepotentialtoobserveleptonnumberviolation(LNV)indisplacedvertexsearchesforheavyneutralleptons(HNL...

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Distinguishing Dirac and Majorana Heavy Neutrinos at Lepton Colliders

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