Heavy Neutral Leptons During The Big Bang Nucleosynthesis Epoch

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Heavy Neutral Leptons During The

Big Bang Nucleosynthesis Epoch

Instituut-Lorentz for Theoretical Physics LEIDEN UNIVERSITY

Thesis submitted in partial fulfillment of the requirements for the degree of

MASTER OF SCIENCE

in PHYSICS

Author : Nashwan Sabti

Student ID : 1367110

Supervisor : Alexey Boyarsky 2nd corrector : Ana Achúcarro

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Heavy Neutral Leptons During The

Big Bang Nucleosynthesis Epoch

Nashwan Sabti

Instituut-Lorentz, Leiden University P.O. Box 9500, 2300 RA Leiden, The Netherlands

July, 2018

Abstract

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T

ABLE OF

C

ONTENTS

1 Introduction 1

1.1 Relevance of this work . . . 1

1.2 Overview . . . 2

2 The Standard Case 3 2.1 Defining the system . . . 3

2.1.1 The background . . . 4

2.1.2 Nuclei . . . 5

2.2 Regimes of particles . . . 7

2.3 Boltzmann equations in the SM . . . 7

2.4 Relevant interactions in the SM . . . 8

2.4.1 Four-particle collision integral . . . 9

2.5 Temperature evolution . . . 10

2.6 System of equations . . . 10

3 Introducing Heavy Neutral Leptons 11 3.1 The type I seesaw Lagrangian . . . 11

3.2 Properties of HNLs . . . 12

3.2.1 Roles of the individual HNLs . . . 13

3.2.2 Dirac spinor from two Majorana spinors . . . 13

3.2.3 Shortcut for calculation matrix elements of HNLs . . . 14

3.2.4 Decoupling temperature . . . 14

3.2.5 Dependence of mixing angle on temperature . . . 15

3.2.6 Contribution to energy density of the Universe . . . 16

3.2.7 Contribution to temperature evolution . . . 16

3.3 Relevant interactions of HNLs . . . 17

3.3.1 Interactions above QCD-scale . . . 17

3.3.2 Interactions below QCD-scale . . . 17

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TABLE OF CONTENTS

3.4 Boltzmann equation for HNLs . . . 21

3.4.1 Three-particle collision integral . . . 21

3.5 Influence of HNLs on BBN . . . 22

4 Results 23 4.1 Simulating BBN withPYBBN . . . 23

4.2 Results for Standard Model BBN . . . 24

4.3 Results forνMSM . . . 26

5 Discussion and Prospects 27 Acknowledgements 29 A Relevant Matrix Elements 31 A.1 Matrix elements in the SM . . . 33

A.1.1 Four-particle processes with leptons . . . 33

A.1.2 Three-particle and four-particle meson decays . . . 34

A.2 Matrix elements for HNLs aboveΛQCD . . . 35

A.2.1 Four-particle processes with leptons only . . . 35

A.2.2 Four-particle processes with leptons and quarks . . . 37

A.3 Matrix elements for HNLs belowΛQCD . . . 38

A.3.1 Three-particle processes with single mesons . . . 38

B Collision Integrals 39 B.1 Three-particle collision integral . . . 39

B.1.1 Case y16=0 . . . 40

B.1.2 Case y1=0 . . . 41

B.2 Four-particle collision integral . . . 42

B.2.1 Case y16=0 . . . 42

B.2.2 Case y1=0 . . . 45

C Temperature Evolution 47

D Neutrino Oscillations 49

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NTRODUCTION

1.1

Relevance of this work

A variety of cosmological, astrophysical and collider based probes can be used to put bounds on parameters of HNLs (Figure1.1). Since the lifetime of an HNL is inversely proportional to the square of the mixing angle, a smaller mixing angle will lead to a longer lifetime, which means that such particles could be present at times relevant for BBN processes. An addition of HNLs to the Universe affects its cosmological expansion and the particle physics processes within compared to the standard case. Therefore, BBN provides a suitable probe for HNLs and gives a lower bound in the aforementioned parameter space.

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CHAPTER 1. INTRODUCTION

FIGURE 1.1: Current limits on the mixing between muon neutrino and

a single HNL up to mass of 500 GeV. The brown region labeled ‘Seesaw’ corresponds to a naive estimate of the mixing scale in the canonical seesaw. The grey region labeled ‘BBN’ corresponds to an HNL lifetime>1 s, which is disfavored by BBN. Figure from [6].

1.2

Overview

This work will be more of a review that highlights the relevant points in this topic, rather than an extensive, repetitive study of what has come before. The interested reader is referred to [9] for a more deep dive into the subject of BBN.

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TANDARD

C

ASE

The origin of primordial elements goes back to the early Universe. When going backwards in time, the Universe can be considered as a shrinking volume with increasing temperature.Naively, once this temperature exceeds the binding energy of a particle, be it element or hadron, it will be destroyed into its constituents. At some temperature one therefore expects a plasma of elementary particles that is in thermodynamic equilibrium. Now, going forward from this point in time, the Universe will be expanding and cooling down. This is the starting point of this chapter.

2.1

Defining the system

The main system is a plasma in an expanding universe. This system can be divided into two subsystems, based on the following processes:

• Cosmology & particle physics (denoted asbackgroundphysics)

• Nuclear physics

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CHAPTER 2. THE STANDARD CASE

2.1.1

The background

At temperatures below QCD-scale,T<ΛQCD∼150 MeV [1], the dominant compo-nents of the background plasma are photons, charged leptons and active neutrinos, with here and there some traces of neutrons and protons. The expansion is gov-erned by the energy density of the plasma and is described by the Friedmann equation

H2

µ

˙

a

a

¶2

=8πG

3 ρ, (2.1)

withH the Hubble parameter,athe scale factor andρthe energy density. In the case of radiation domination, this equation can be written as

H=1.66

pg ?

Mpl T

2, (2.2)

with g? the effective number of relativistic degrees of freedom andMpl the Planck mass.

The background is assumed to be homogeneous and isotropic. The energy density

ρtogether with the total pressureP then satisfy the energy conservation law dρ

dt+3H(ρ+P)=0. (2.3)

While the Universe is expanding and cooling down, some species can get out of equilibrium (also known asdecoupling). This happens at a temperature defined when the inequality

Γ(T) < H(T) (2.4) starts to hold. Here Γ(T) is the rate of the reaction that keeps the particle in equilibrium. Relevant events during cooldown are [22]:

1.5 MeV

νdecoupling

0.8 MeV n decoupling

0.5 MeV

e±annihilation

T

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2.1. DEFINING THE SYSTEM

• Baryon chemical decoupling at T 0.8 MeV. This value is estimated by considering the reactions

np+e−+ν¯e (2.5)

n+νep+e− (2.6)

n+e+↔p+ν¯e (2.7)

• Electron-positron annihilation atT≈0.5 MeV.

Temperatures higher thanΛQCD are not considered in the SM, since before neu-trino decoupling the plasma is in equilibrium and nothing of importance for BBN happens.

2.1.2

Nuclei

Once neutrons decouple aroundT0.8 MeV, they will decay according to Eq.2.5. This will go on until production of deuterium starts through the reaction

n+pD+γ (2.8)

Due to the small baryon-to-photon ratio, the production of deuterium will not start at temperatures close to its binding energy∆D≈2.2 MeV. Indeed, at such temperatures there are many photons with energies higher than∆D, which will destroy the deuterium nuclei immediately after they are created. This is also known as thedeuterium bottleneck. Production of deuterium will become effective at temperatures much lower:

nbaryons=ηBnγ=nγ(E>∆D) (2.9) Solving this equation gives an estimate for the temperature of start of BBN,

TBBN70 keV.

Once deuterium is formed, a series of nuclear reactions will lead to production of mainly4He. Heavier nuclei are produced in much smaller abundances, because

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CHAPTER 2. THE STANDARD CASE

• the lack of sufficient densities of lighter nuclei diminishes the rates of nuclear reactions that produce heavier nuclei. Such processes are on the verge of decoupling around BBN temperatures.

• the Coulomb barrier causes electrostatic repulsion.

The last point is the main reason for decoupling of reactions involving heavier nuclei. Therefore,4He will be the most abundant element created during BBN, with here and there traces of heavier elements [9].

Following the previous discussion, the observables of BBN are abundances of light elements. In the Standard Model the only free parameter is the baryon-to-photon ratioηB [9]. Thus, measuring this quantity (by abundance measurements or CMB measurements) leads to predictions of all primordial element abundances.

The most relevant abundance is that of4He, which can be described by the mass fractionY4He= 2

nn n p

¯ ¯T

BBN 1+nnn p¯¯

TBBN

. This quantity is determined by:

• Neutron-to-proton ratio nn

np at time of neutron decoupling. Using equilibrium

physics:

nn

np

¯ ¯ ¯ ¯

Tndec

=(mnTndec)

3/2e−mn/Tndec (mpTndec)3/2e−mp/Tndec ≈

e−(mn−mp)/Tndec (2.10)

Tndeccan be determined by equating the weak reaction rate in Fermi theory,

Γ∼G2

FT

5, to the Hubble rate in Equation2.2, which gives

Tndec0.8 MeV⇒ nn

np

(Tndec)≈0.16

• Time ∆t between neutron decoupling and start of deuterium production. During this period neutrons will decay with mean lifetimeτ:

nn

np

¯ ¯ ¯ ¯

TBBN ≈

³

nn

np(Tndec)

´

e−∆t/τ

1+nn

np(Tndec)

£

1−e−∆t/τ¤ (2.11)

∆tcan be obtained from Equation2.2, which gives∆t150 s. Plugging this in the equation above gives nn

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2.2. REGIMES OF PARTICLES

Important quantities to keep in mind when studying BSM physics are therefore:

• Neutron-to-proton ratio at time of neutron decoupling.

• Hubble parameterH.

Chapter4will elaborate a little more on computations involving nuclei.

2.2

Regimes of particles

It is useful to categorize particles in the plasma in the following way:

• Massless & in-equilibrium (photons)

• Massive & in-equilibrium (charged leptons)

• Massless & out-of-equilibrium (active neutrinos)

• Massive & out-of-equilibrium (protons, neutrons and nuclei)

The first two groups of particles can be treated using equilibrium expressions. Using equilibrium physics for particles in the latter two groups gives incorrect abundances of heavy elements [20]. Instead, distributions of particles in the latter two groups are prone to distortions due to interactions around decoupling. These particles must therefore be treated properly within the framework of the Boltzmann equation.

2.3

Boltzmann equations in the SM

For the time being, only active neutrinos will be considered here. The baryons and nuclei will be dealt with in Chapter4. Throughout this work no lepton asymmetry is assumed. At temperatures higher than the neutrino decoupling temperature, neutrinos will have a Fermi-Dirac distribution. At lower temperatures, a set of three Boltzmann equations must be solved [25]:

dfνα

dt =H

dfνα

d lna=

X

β

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CHAPTER 2. THE STANDARD CASE

withα,β∈{e,µ,τ},Pβα time averaged transition probabilities andIα the collision term, which encodes the details of interactions. The termsPβαaccount for neutrino oscillations; more details and expressions are given in AppendixD. The collision term Iα for particleνα in the reaction

να+2+3+...+K⇐⇒(K+1)+(K+2)+...+Q

has the general form [20]

Iα= 1 2gαEα

X

in,out

Z Q Y

i=2

d3pi

(2π)32E i

S|M|2F[f](2π)4δ4(Pin−Pout), (2.13)

with gα the degrees of freedom of να, ‘in’ the initial states {να, 2, ...,K}, ‘out’ the final states {(K+1), (K+2), ...,Q}, S the symmetry factor, |M|2 theunaveraged, squared matrix element summed over helicities of initial and final states andF[f] the functional describing the particle population of the medium, given by

F[f]=(1±fνα)...(1±fK)fK+1...fQfνα...fK(1±fK+1)...(1±fQ). (2.14)

Here (1−f) is the Pauli blocking factor used for fermions and (1+f) the Bose enhancement factor used for bosons. The sum in Eq.2.13 runs over all possible initial and final states involvingνα.

2.4

Relevant interactions in the SM

All relevant reactions with active neutrinos should be considered:

• Neutrino pair annihilation into neutrino pair and neutrino-neutrino scatter-ing:ν+νν+νandν+νν+ν

• Neutrino-charged lepton scattering:ν+`±⇔ν+`±

• Neutrino pair annihilation into charged lepton pair and vice versa:

ν+ν`±+`

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2.4. RELEVANT INTERACTIONS IN THE SM

FIGURE2.1: Leading order diagrams contributing to the collision integral in the Boltzmann equation for active neutrinos.

The squared matrix element in Fermi theory is of the following form for all reactions considered:

|M|2= X i6=j6=k6=l

£

K1(pi·pj)(pk·pl)+K2mimj(pk·pl

, (2.15)

withK1, K2constants. Expressions for|M|2of all relevant SM interactions involv-ing active neutrinos are given in TableA.1. Since the baryon-to-photon ratio is very small, the assumption is made that Eqs.2.5-2.7do not alter the electron-neutrino distribution and are therefore neglected in the collision integral.

2.4.1

Four-particle collision integral

In the case of four-particle interactions, like

να+2⇔3+4

and similar crossing processes, the nine-dimensional collision integral can be reduced to a two-dimensional one. Using Eq.2.15in2.13, the collision integral for a single reaction will be of the form

Iα,single= 1

64π3g

αEαpα

Z

dp2dp3p2p3

E2E3SF[f]D(pνα,p2,p3,p4)θ(E4−m4), (2.16)

where the D-function is a conditional polynomial function of the four momenta

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CHAPTER 2. THE STANDARD CASE

2.5

Temperature evolution

The conservation of energy equation2.3can be used to derive an equation for the temperature evolution of the plasma. Usingρ=ρeq+ρnoneq(energy densities of par-ticles in (non-)equilibrium) in Eq.2.3together with the fact that the temperature of the plasma is only defined for particles in equilibrium gives

µdρ

eq dT

dT

dt +

dρnoneq dt

= −3H¡

ρ+P¢

dT

dt = −

3H(ρ+P)+dρnoneqdt dρeq

dT

. (2.17)

Expressions forρeq andρnoneq can be substituted in this equation to obtain an explicit formula for the temperature evolution. This is done in AppendixC.

2.6

System of equations

There are five equations that describe the primordial plasma and which have to be solved:

• Three Boltzmann equations for active neutrinos

• Friedmann equation

• Temperature evolution equation

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3

I

NTRODUCING

H

EAVY

N

EUTRAL

L

EPTONS

Over the last few years a plethora of extensions to the SM have been proposed to solve the yet unexplained observations of dark matter, baryon asymmetry of the Universe and neutrino masses. One of those ambitious extensions that aims to take down all of these problems at once is the νMSM. Introduced in [2, 3], this model adds three Heavy Neutral Leptons (HNLs) to the SM, making them the right-handed counterparts of active neutrinos. This chapter will explore the possibilities that BBN offers in constraining some of the parameters in this model.

3.1

The type I seesaw Lagrangian

The most general renormalizable Lagrangian that includes three right-handed neutrinos to the SM Lagrangian has the form [7]

L =LSM+iNI∂µγµNI−

µ

FαILαφeNI+

MI 2 N

c

INI+h.c.

, (3.1)

with I ={1, 2, 3}, α={e,µ,τ}, NI right-handed neutrinos (also known as HNLs or sterile neutrinos), FαI Yukawa couplings, MI HNL Majorana masses,Lα the SM lepton doublet andφe≡iσ2φ∗ the conjugated Higgs doublet. Without loss of

generality, the Majorana mass matrix can be chosen diagonal.

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CHAPTER 3. INTRODUCING HEAVY NEUTRAL LEPTONS

The mass terms can then be rewritten as

Lmass= − 1 2

³

ν Nc´

Ã

0 MD MDT MI

! Ã

νc

N

!

+h.c. , (3.2)

whereN=(N1 N2 N3)T,ννeνµντ

¢T

andc indicates charge conjugation. Block-diagonalizing the mass matrix and assumingMD<<MI yields the type 1 seesaw formula

(Mν)αβ= −X I

(MD)αIMI−1(MDT)Iβ, (3.3) whereMν is the 3×3 active neutrino mass matrix, which can be diagonalized by a unitary transformation with the PMNS-matrixVPMNS. Defining the mixing angle matrix as

θαI≡(MD)αIM−I1, (3.4) the charge eigenstates ν and N in Eq. 3.2can be written in terms of the mass eigenstatesνmandNmof the matrix diag(VPMNST MνVPMNS,MI) up to leading order inθ:

ν=VPMNSνm+θNmc (3.5)

N=Nmθ†VPMNSνmc . (3.6) The light mass eigenstates νm almost correspond to active neutrinos, while the heavy mass eigenstatesNmalmost correspond to sterile neutrinos. The expression forνcan then be substituted in the weak interaction part of the SM Lagrangian, which gives the interaction of HNLs with SM particles:

LHNL,int= −

g

4 cosθW

ZµX

I

X

α(

Nmc)Iθ∗Iαγµ(1−γ5)να

− g 2p2W

+

µ

X

I

X

α

(Nmc)IθIαγµ(1−γ5)`α+h.c. (3.7) Therefore, HNLs couple to SM fields in a similar way as active neutrinos, but with an additional mixing angle that suppresses the interaction.

3.2

Properties of HNLs

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3.2. PROPERTIES OF HNLS

3.2.1

Roles of the individual HNLs

Among the three HNLs, N1 serves as a dark matter candidate, which means it must be light, stable and very weakly interacting. In this scenario, the Yukawa coupling constantsFα1 in Eq.3.1are required to be very small. As a result, this particle gives a negligible contribution to the active neutrino mass matrix (Eq.3.3) as well as the baryon asymmetry of the Universe [7]. Therefore, only N2 andN3

are responsible for these two phenomena andN1will be neglected from this point forward. In order to achieve successful baryogenesis, the masses ofN2andN3are required to be close to degenerate.

3.2.2

Dirac spinor from two Majorana spinors

For the sake of convenience, bothN2andN3are assumed to have the same mass

MN and same set of mixing angles {θe,θµ,θτ}. Since both spinors are two-component Majorana spinors, it is possible to combine them intooneDirac spinor with four degrees of freedom. The HNL mass terms are approximately given by

LHNL,mass≈ − 1 2MN

³

N2cN2+N3cN3+h.c.

´

. (3.8)

Constructing two Majorana spinorsχandξout ofN2andN3,

χ=N2c+N2 (3.9)

ξ=N3c+N3, (3.10) and substituting this in Eq.3.8gives

LHNL,mass= − 1 2MN

³

χχ+ξξ

´

. (3.11)

Define the Dirac spinor as

ND=p1 2

¡

χ+iξ¢ (3.12)

NDc =p1 2

¡

χ−iξ¢ (3.13)

and Eq.3.11becomes

LHNL,mass= − 1 2MN

³

NDND+Nc

DN c D

´

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CHAPTER 3. INTRODUCING HEAVY NEUTRAL LEPTONS

3.2.3

Shortcut for calculation matrix elements of HNLs

Expressions forNIcin terms of NDandNDc,

N2+N2c=p1

2

¡

ND+ND

(3.15)

N3+N3c= 1 ip2

¡

NDND

, (3.16)

together withθ2α=θ3α=θα can be substituted in Eq.3.7to give

LHNL,int= −

g

4 cosθW

ZµX

α 1 p

2(ND+N c

D−iND+iN c D)θ

αγµ(1−γ5)να − g

2p2W

+ µ X α 1 p

2(ND+N c

D−iND+iN c D)Iθ

αγµ(1−γ5)`α+h.c. = − g

4 cosθW

ZµX

α(

e−iπ4ND+eiπ4Nc

D)θαγµ(1−γ5)να − g

2p2W

+

µ

X

α

(e−iπ4ND+eiπ4Nc D)Iθ

αγµ(1−γ5)`α+h.c. = − g

4 cosθW

ZµX

α(

ND+Nc

D)θ

αγµ(1−γ5)να − g

2p2W

+

µ

X

α

(ND+Nc

D)Iθαγµ(1−γ5)`α+h.c. , (3.17) where thee±iπ4 are absorbed in theND fields. It can be seen now that the charac-teristics of the Majorana particles have not disappeared: particle and its charge conjugate still interact in the same way. Since no lepton asymmetry is assumed, both particle and its charge conjugate can be treated on an equal footing. Therefore, it is enough to compute only one matrix element, which can be obtained by

|M|2N= |θα|2|M|2ν (3.18) The effect of the charge conjugated particle is then taken into account through the degrees of freedom. As long as the Dirac particle has the same lifetime, mixing pattern and spectrum as the Majorana particle, the two cases are equivalent.

3.2.4

Decoupling temperature

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3.2. PROPERTIES OF HNLS

A naive approach for determining the decoupling temperature is to equate the weak interaction rate in Fermi theory in the ultra-relativistic limit to the Hubble rate for a radiation dominated universe:

1.66pg Mpl T

2

=θM2(T)G2FT5, (3.19) whereθM(T) is the effective mixing angle in a medium, which arises from a self-energy term induced by interactions of active neutrinos with particles in the medium. An explicit expression is given by [5,7,23]

θ2 M(T)=

θ2

µ

1+M2p2

N

µ

16G2 F

αW p

¡

2+cos2θ W

¢7πT4

360

¶¶2

+θ2

, (3.20)

withθthe mixing angle in vacuum andαW the weak coupling constant. Plugging θ2 =10−6, g

∗=20 and p∼πT in Eq. 3.19 yields a decoupling temperature of

Tdec160 MeV for MN100 MeV. For higher masses or larger mixing angles this approach becomes less credible, because at some point the HNLs will decouple non-relativistically.

More sophisticated approaches have been developed in [15, 16, 27] and found decoupling temperatures ofO(1) GeV for masses 0.5≤MN1 GeV. It has been shown in [15] that HNLs with such masses will always enter equilibrium at temperatures aboveT5 GeV.

This work will use these results as a guide by looking at what temperatures the distribution function will start to differ from the equilibrium one. This transition point should give an estimation for the decoupling temperature.

3.2.5

Dependence of mixing angle on temperature

As can be seen in the previous subsection, the mixing angle θM depends on the temperature of the medium. However, it can be shown thatθMdiffers fromθ as [25]

θM−θ θ

GFT6 M2

WM 2 N

∼10−11

µ T

100 MeV

¶6µ

10 MeV

MN

¶2

, (3.21)

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3.2.6

Contribution to energy density of the Universe

The inclusion of an HNL to the background described in subsection 2.1.1 has consequences for its expansion. Assuming the HNL decouples non-relativistically, its energy density at temperaturesT<Tdec<TNRMis given by

ρN(T)∼MNnN(Tdec)

³adec

a

´3

∼MNnN(Tdec) g∗(T)T 3

g(Tdec)T3

dec ∼4

µM

NTdec 2π

¶32

e−MN/Tdec g∗(T)T 3

g(Tdec)T3

dec

, (3.22)

where entropy conservation, g(T1)a31T31 = g(T2)a32T23, is used. Equating this density to the energy density of radiation gives

4

µM

NTdec 2π

¶32

e−MN/Tdec g∗(T)T 3

g(Tdec)T3

dec

=g(T)π 2 30T

4

=⇒T30 keV (3.23)

for MN50 MeV,θ2∼10−4,Tdec30 MeV and g(Tdec)∼10.

This means that the Universe will become matter dominated at the time nu-clear reactions are taking place. Therefore, if the HNL has not decayed yet long before this time, it could have a direct impact on the abundances of primordial elements.

3.2.7

Contribution to temperature evolution

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3.3. RELEVANT INTERACTIONS OF HNLS

3.3

Relevant interactions of HNLs

All interactions of HNLs with SM particles are mediated by charged and neutral currents. The higher the mass, the more channels will be available. All relevant interactions together with the corresponding matrix elements for HNL masses up to∼1 GeV are summarized inA.2andA.3.

Since HNLs can decouple at temperatures higher thanΛQCD∼150 MeV, interac-tions above and below this border must be distinguished.

3.3.1

Interactions above QCD-scale

At temperaturesT>ΛQCDno bound states of quarks exist and only two types of reactions should be considered:

• Interactions with active neutrinos and charged leptons.

• Interactions with free quarks. Here, only interactions with two quarks are considered, since multiquark final states are suppressed by higher orders of the couplingαsin perturbative QCD [8].

Contributing diagrams are of the following form:

FIGURE3.1: Leading order diagrams contributing to the collision integral in the Boltzmann equation for an HNL at temperaturesT>ΛQCD.

3.3.2

Interactions below QCD-scale

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four-CHAPTER 3. INTRODUCING HEAVY NEUTRAL LEPTONS

particle interactions are considered:

• Interactions with active neutrinos and charged leptons like before.

• Interactions with a single meson in the final state. Only HNL decays are considered here, since the creation of a mesonhin the reactionN+v/`→h

is only possible whenMN<Mh. Moreover, at temperatures ofO(1) MeV the rate of this reaction is much smaller than the decay rate of HNLs. The only contributing diagram is therefore:

FIGURE 3.2: Leading order diagram for three-particle reactions con-tributing to the collision integral in the Boltzmann equation for an HNL at temperaturesT<ΛQCD.

The branching ratios of the relevant HNL decay channels are plotted in Figure3.3.

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3.3. RELEVANT INTERACTIONS OF HNLS

Interactions with more than two mesons in the final state are not considered, except for reactions with two pions in the final states, such asNν+π++π−. These reac-tions are taken into account by resonance ofρmeson, i.e.,Nν+ρ0ν+π++π−. It has been shown in [6] that the decay width of both two reactions coincide and, therefore, decay into two pions happens predominantly via decay intoρmeson. The decay channel to two pions is also open for 2mπ<MN<mρ, but this contribution is negligible.

The contribution of decays into multi-meson final states to the full hadronic decay width can be estimated by comparing the combined decay width of single-meson final states with the full hadronic decay width. The full hadronic decay width can be estimated by considering the total decay width into quarks. The result can be seen in Figure3.4. Multi-meson final states become important for masses MN>1 GeV, while for masses smaller it is enough to consider only single-meson channels as the hadronic decay modes.

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CHAPTER 3. INTRODUCING HEAVY NEUTRAL LEPTONS

3.3.3

Interactions of unstable decay products with plasma

HNLs with masses MN <105 MeV will decay into stable particles. HNLs with higher masses will have decay products that are unstable. Some of these unstable decay products will interact with the plasma before they decay. The analysis here will be done for muons, but can be applied to all other particles as well.

There are three important events to consider:

1. µ± is created from HNL decay

The distribution function of these muons is a non-thermal distribution fnoneq. 2. µ± thermalizes

The muon-photon scattering rate is higher than the muon decay rate:

Γγµα

2 mµEγT

3

γ ∼10−9 MeV vs. Γµ,decay ∼10−16 MeV. This means that the muons will release their energy into the plasma and equilibrate before they decay. This process increases the temperature of the plasma and makes the muons non-relativistic. After thermalization the muons will share the same temperature as the plasma and will have a thermal distribution

fthermal =e−mµT−µe− p2

2mµT, where µ is determined by the condition that the

number density before and after thermalization must be equal. The collision term corresponding to this process is then estimated as

Ithermalization fthermal−fnoneq

∆t ,

with∆tthe timestep of the simulation. The same procedure is followed for charged pions and charged kaons.

3. µ± decays

The main decay channel of muons isµe+νe+ν

µ. The muon has a lifetime,

τµ∼10−6 sec., that is much smaller than the timestep of the simulation. This poses a problem right away: when the evolution of the distribution function for the muon and active neutrinos is computed as∆f = Icoll∆t, the behaviour ofIcollis not resolved. It is assumed to be constant during the whole timestep

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3.4. BOLTZMANN EQUATION FOR HNLS

This issue can be solved by using dynamical equilibrium. Consider the chain

HNL +∆ µ ν

N ∆N

The timestep∆tis much smaller than the lifetime of the HNL, which means that there is approximately a constant inflow of muonsduringeach timestep. Since the amount of muons created∆Ndecays almost instantaneously, the same number of active neutrinos is created: for each muon that decays, one electron neutrino and one muon neutrino is created. Now a scalingαcan be introduced in∆f =Icoll∆tαsuch thatR

d3p∆f/(2π)3=∆N.

3.4

Boltzmann equation for HNLs

In this model, one more equation must be added to the system of equations in Section2.6. At temperatures below the HNL decoupling temperature, the HNL distribution function fN can be obtained by solving the Boltzmann equation

dfN

dt =IN , (3.24)

with IN the collision term. Besides the four-particle collision integral in Eq.2.16

there is also a three-particle collision integral.

3.4.1

Three-particle collision integral

In the case of a two-body decay

N→2+3, (3.25)

the six-dimensional collision integral in Eq.2.13can be reduced to a one-dimensional integral. Since the matrix element for three-particle interactions does not depend on the four-momenta of the particles, it has the simple form

IN,single= πS|M|

2 4gNENpN

Z

dp2p2

E2F(f)Xθ

µ

(gEN−Ef2)2−

x2 M2m

2 3

(3.26)

for each reaction taken into account. Here,X is given by

X =π

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CHAPTER 3. INTRODUCING HEAVY NEUTRAL LEPTONS

3.5

Influence of HNLs on BBN

Subsection2.1.2listed the two main points that determine the course of BBN. If HNLs decay before neutrino decoupling, the plasma will just re-equilibrate and nothing will change. But if decay happens after neutrino decoupling, then HNLs can influence BBN by

• their contribution to the cosmological energy density.

An increase in the energy density leads to a higher expansion rate according to the Friedmann equation2.1. The decoupling temperature of weak inter-actions, determined by H=Γ∼Tdec5 , will then be higher compared to the standard case. Therefore, the nn

np−ratio at time of neutron decoupling, Eq.

2.10, will be larger. This will lead to a larger helium mass fractionY4He. • their decay products.

HNL decay injects active neutrinos and electrons/positrons into the plasma with energies that may be different from typical energies of plasma particles.

Decay intoνµandντwill increase the expansion rate with respect to the standard case and this will therefore increase the nn

np−ratio at neutron decoupling and thus increaseY4He.

Decay into νe has two effects: (I) it will increase the expansion rate similarly to above and (II) it will preserve equilibrium between neutrons and protons for a longer time, since they participate in the reactions in Eqs.2.5-2.7. The latter effect would make the nn

np−ratio at neutron decoupling smaller. Effect (II) is stronger than (I) [10] and the net effect will be a decrease ofY4He.

Decay into e±will inject more energy into electromagnetic part of the plasma and heat it up, which increases the expansion rate and increases

Y4He.

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C

H

A

P

T

E

R

4

R

ESULTS

This chapter will present results for neutrino decoupling spectra in the SM and bounds on HNL lifetime as a function of its mass. The latter is done for HNL masses up toMN=100 MeV where mixing with only electron neutrino is turned on. The bounds are obtained by comparing the4He mass fraction obtained from simulations with measurements done in [4].

4.1

Simulating BBN with

PY

BBN

The computational scheme of the code (PYBBN) used to do the simulations will be

briefly summarized here. An extensive user guide with schemes, approximations, results and comparisons with literature will be available soon on the homepage1of the code.

Simulations are done in two steps: (I) The background physics and the rates of the reactions in Eqs.2.5-2.7are computed inPYBBN. This involves solving the system of equations in Section2.6for the evolution of temperature, scale factor and distribution functions of decoupled species. (II) The cosmological quantities

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CHAPTER 4. RESULTS

together with the aforementioned rates are tabulated and passed to an external code, the modified KAWANO code [25], that takes care of the nuclear part of the simulation and outputs the light element abundances.

The4He abundance can then be compared with what is determined by [4] as

Y4He=0.2452−0.2696 (2σinterval) (4.1)

4.2

Results for Standard Model BBN

The temperature evolution is the first interesting point to consider, since it will show the heat up of the plasma due to electron-positron annihilation. The result is shown in Figure4.1.

FIGURE 4.1: The photon temperature divided by active neutrino tem-perature. The increase here is due to electron-positron annihilation into photons. Dashed curve is from [25].

The theoretical value is obtained by using entropy conservation:

Tγ

Tν = aTγ

aTν =

µg

∗(Tbefore)

g(Tafter)

¶13

=

Ã7

8·4+2 2

!13

(31)

4.2. RESULTS FOR STANDARD MODEL BBN

The next step is to compare the active neutrino spectra at the end of electron-positron annihilation with their equilibrium distribution at high temperature. The result is shown in Figure4.2.

FIGURE4.2: Ratio of active neutrino decoupled spectra to their equilibrium distribution before the onset of BBN. The upper curves show the distortion of the electron neutrino spectrum and the lower of muon and tau neutrinos. Neutrino flavour oscillations are not taken into account here. Dashed curves are from [25], dotted from [21].

In Fermi theory the cross section increases with momentum asσ∝G2Fp2, which means that neutrinos with higher momenta stay longer in equilibrium. Since these neutrinos decouple later, they will briefly experience the heat-up of the plasma due to electron-positron annihilation, shown in Figure4.1, and the corresponding increase inaT. Their equilibrium distribution function, given by

fν= 1 eTp+1

= 1

eaTy +1

,

increases then accordingly.

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CHAPTER 4. RESULTS

plasma or to be created from muon and tau neutrinos. The cross section of electron neutrinos is therefore larger and they stay longer in equilibrium.

When the relevant cosmological quantities are passed to the modified KAWANO code, a value ofY4He=0.24793 is obtained; in agreement with the result in Eq.4.1.

4.3

Results for

ν

MSM

Results are shown here for HNLs with masses up to 100 MeV that mix with electron neutrino only. There are four channels through which the HNL can decay,

Nνe+νe+νe Nνe+νµ+νµ Nνe+ντ+ντ Nνe+e++e− , and from which the lifetime can be computed as

τN=Γ−N1=

192π3

G2

F|θe| 2M5

N

¡1

4

¡

1+4 sin2θW+8 sin4θW

¢

+1¢ (4.3)

This equation is used to obtain the plot in Figure4.3.

FIGURE4.3: Bounds on the lifetime of HNLs that mix only with electron

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C

H

A

P

T

E

R

5

D

ISCUSSION AND

P

ROSPECTS

The results in Figures 4.1,4.2 and4.3are all consistent with the literature. A number of deviations have been found between the code used in this work and the code of [25], but these do not affect the bounds significantly. All these deviations will be included in the user guide of pyBBN.

The bounds obtained in this work are for HNLs that mix only with electron neutrinos. In general, the mixing pattern can be random. However, it turns out that for massesMN up to∼105 MeV the mixing pattern is not very important, as can be seen in Figures 4, 5 and 6 of [25]. Mixing with muon neutrino only changes the lifetime bound subtly. Once neutrino oscillations are included, the mixing pattern becomes even less important. Therefore, it is the amount of energy injected in the plasma that is more relevant, which depends on the lifetime and mass of the HNL. For higher masses the contribution of HNLs to the expansion rate will probably be an increasingly important factor, since they will dominate the energy density at some point if they have not decayed yet. It is unknown at the moment what holds for masses exceeding 105 MeV, because the decay products themselves will be unstable. For high masses one can therefore expect a shower of decay products, each of which will influence the course of BBN in its own way.

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(35)

A

CKNOWLEDGEMENTS

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(37)

A

P

P

E

N

D

I

X

A

R

ELEVANT

M

ATRIX

E

LEMENTS

The matrix elements listed here are not averaged over any helicities. Subsection

A.1contains the reactions involving SM particles only, SubsectionA.2the reactions involving HNLs above QCD-scale and SubsectionA.3the reactions involving HNLs below QCD-scale. HNL decay channels with a branching ratio of at least 1% for some mass below∼1 GeV are considered in this work (see FigureA.1). The results for HNLs do not take into account charge conjugated channels, which are possible if they are Majorana particles.

The explicit determination of matrix elements involving multiple mesons can be extremely challenging. Therefore, an approximation has been used by assuming the matrix element to be constant and using the definition of decay width,

Γ= 1 2gM

Z Ã Y

i

d3yi

(2π)32E i

!

|M|2(2π)4δ4(P−X i

Pi),

together with its measured value (from e.g. [24]) to solve for|M|2. For three-particle reactions this method gives the exact matrix element.

The values of the meson decay constants used in Subsection A.3 are from [6] and summarized below.

fπ0 fπ± fη fρ0 fρ± fω fη0 fφ

(38)

APPENDIX A. RELEVANT MATRIX ELEMENTS

FIGURE A.1: List of HNL decay channels with branching ratios more than

(39)

A.1. MATRIX ELEMENTS IN THE SM

A.1

Matrix elements in the SM

A.1.1

Four-particle processes with leptons

Process (1+2→3+4) S SG2Fa−4|M|2

να+νβνα+νβ 1 32 (Y1·Y2) (Y3·Y4) να+νβνα+νβ 1 32 (Y1·Y4) (Y2·Y3) να+νανα+να 12 64 (Y1·Y2) (Y3·Y4) να+νανα+να 1 128 (Y1·Y4) (Y2·Y3) να+νανβ+νβ 1 32 (Y1·Y4) (Y3·Y2)

νe+νe→e++e− 1 128

h

g2L(Y1·Y3) (Y2·Y4)+g2R(Y1·Y4) (Y2·Y3) +gLgRa2m2e(Y1·Y2)i

νe+e−→νe+e− 1 128

h

g2

L(Y1·Y2) (Y3·Y4)+g 2

R(Y1·Y4) (Y3·Y2) −gLgRa2m2e(Y1·Y3)

i

νe+e+→νe+e+ 1 128

h

g2L(Y1·Y4) (Y3·Y2)+g2R(Y1·Y2) (Y3·Y4) −gLgRa2m2e(Y1·Y3)

i

νµ/τ+νµ/τ→e++e− 1 128

h f

gL2(Y1·Y3) (Y2·Y4)+g2R(Y1·Y4) (Y2·Y3) +gfLgRa

2m2

e(Y1·Y2)

i

νµ/τ+e−→νµ/τ+e− 1 128

h f

gL2(Y1·Y2) (Y3·Y4)+g2R(Y1·Y4) (Y3·Y2) −gfLgRa

2m2

e(Y1·Y3)

i

νµ/τ+e+→νµ/τ+e+ 1 128

h f

gL2(Y1·Y4) (Y3·Y2)+g2

R(Y1·Y2) (Y3·Y4) −gfLgRa

2m2

e(Y1·Y3)

i

TABLEA.1: Squared matrix elements for weak processes involving active

neutrinos and electrons/positrons. S is the symmetry factor and α,β∈ {e,µ,τ}, whereα6=β. Here:gR=sin2θW, gL=1/2+sin2θWandgfL= −1/2+

(40)

APPENDIX A. RELEVANT MATRIX ELEMENTS

A.1.2

Three-particle and four-particle meson decays

Process (1→2+3) S |M|2

π0γ+γ 1 α2

emm4π

£

2π2fπ2¤−1

π+µ++ν

µ 1 2G2F|Vud| 2f2

πm4µ

·

m2π m2µ−1

¸

TABLE A.2: Squared matrix elements for pion decays.

Process (1→2+3+4) |M|2

K+π0+e++νe 1.42906·10−13

K+π++π+π+ 1.85537·10−12

K0Lπ±+e∓+νe 2.80345·10−13

K0

L→π±+µ∓+νµ 3.03627·10− 13

K0Lπ0+π0+π0 1.05573·10−12

K0Lπ++π+π0 8.26989·10−13

ηπ0+π0+π0 8.70984·10−2 ηπ++π−+π0 6.90629·10−2

ηπ++π−+γ 4.66530·10−3

ωπ++π−+π0 1.14569·103

η0π++π+η 4.38880·101 η0π0+π0+η 2.00986·101

Process (1→2+3) |M|2 [MeV2]

K+π++π0 3.28177·10−10

K+µ++νµ 8.78918·10−10

K0Sπ++π− 1.53713·10−7

K0

S→π

0+π0 6.71800·10−8 ηγ+γ 1.42174·101

ρ0π++π1.86839·107 ρ+π++π0 1.86390·107 ωπ0+γ 8.55086·104

η0ρ0+γ 8.04463·103 φ→K++K− 1.28798·106

φ→K0

L+K 0

S 1.03471·10 6 φρ0+π0 2.86706·105

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A.2. MATRIX ELEMENTS FOR HNLS ABOVEΛQCD

A.2

Matrix elements for HNLs above

Λ

QCD

A.2.1

Four-particle processes with leptons only

Process (1+2→3+4) S SG−2

F a− 4|M|2

N+νβνα+νβ 1 32|θα|2(Y1·Y2) (Y3·Y4)

N+νβνα+νβ 1 32|θα|2(Y1·Y4) (Y2·Y3)

N+νανα+να 12 64|θα|2(Y1·Y2) (Y3·Y4)

N+νανα+να 1 128|θα|2(Y1·Y4) (Y2·Y3)

N+νανβ+νβ 1 32|θα|2(Y1·Y4) (Y3·Y2)

N+νee++e1 128|θe|2hg2

L(Y1·Y3) (Y2·Y4)+g 2

R(Y1·Y4) (Y2·Y3) +gLgRa2m2e(Y1·Y2)

i

N+e−→νe+e− 1 128|θe|2

h

g2L(Y1·Y2) (Y3·Y4)+g2R(Y1·Y4) (Y3·Y2) −gLgRa2m2e(Y1·Y3)

i

N+e+νe+e+ 1 128|θe|2hg2L(Y1·Y4) (Y3·Y2)+g2R(Y1·Y2) (Y3·Y4) −gLgRa2m2e(Y1·Y3)i

N+νµ/τe++e− 1 128¯¯θµ/τ ¯ ¯

2h f

gL2(Y1·Y3) (Y2·Y4)+g2

R(Y1·Y4) (Y2·Y3) +gfLgRa

2m2

e(Y1·Y2)

i

N+e−ν

µ/τ+e− 1 128¯¯θµ/τ ¯ ¯

2h f

gL2(Y1·Y2) (Y3·Y4)+g2

R(Y1·Y4) (Y3·Y2) −gfLgRa

2m2

e(Y1·Y3)

i

N+e+→νµ/τ+e+ 1 128

¯ ¯θµ/τ

¯ ¯

2h f

gL2(Y1·Y4) (Y3·Y2)+g2R(Y1·Y2) (Y3·Y4) −gfLgRa

2m2

e(Y1·Y3)

i

TABLEA.4: Squared matrix elements for weak processes involving HNLs and leptons. S is the symmetry factor and α,β∈{e,µ,τ}, where α6=β. Here: gR=sin2θW, gL=1/2+sin2θW andgfL= −1/2+sin

2θ

(42)

APPENDIX A. RELEVANT MATRIX ELEMENTS

Process (1+2→3+4) S SG−2

F a− 4|M|2

N+νµe−+µ+ 1 128|θe|2(Y1·Y4) (Y2·Y3)

N+νee++µ1 128¯ ¯θµ

¯ ¯

2

(Y1·Y3) (Y2·Y4)

N+e−νe+µ− 1 128¯¯θµ ¯ ¯

2

(Y1·Y2) (Y3·Y4)

N+e+νµ+µ+ 1 128|θe|2(Y1·Y4) (Y3·Y2)

N+νµµ++µ− 1 128¯¯θµ ¯ ¯

2h

g2L(Y1·Y3) (Y2·Y4)+g2R(Y1·Y4) (Y2·Y3) +gLgRa2m2µ(Y1·Y2)

i

N+νe/τµ++µ− 1 128|θe/τ|2

h f

gL2(Y1·Y3) (Y2·Y4)+g2R(Y1·Y4) (Y2·Y3) +gfLgRa

2m2

µ(Y1·Y2)

i

Process (1→2+3+4) S SG−2

F a− 4|M|2

Nνα+νβ+νβ 1 32|θα|2(Y1·Y4) (Y2·Y3)

N→να+να+να 12 64|θα| 2(Y

1·Y4) (Y2·Y3)

Nνe+e++e− 1 128|θe|2

h

g2L(Y1·Y3) (Y2·Y4)+g2R(Y1·Y4) (Y2·Y3) +gLgRa2m2e(Y1·Y2)i

Nνµ/τ+e++e1 128¯ ¯θµ/τ

¯ ¯

2h f

gL2(Y1·Y3) (Y2·Y4)+g2

R(Y1·Y4) (Y2·Y3) +gfLgRa

2m2

e(Y1·Y2)

i

N→νµ+e−+µ+ 1 128|θe|2(Y1·Y4) (Y2·Y3)

Nνe+e++µ− 1 128¯¯θµ ¯ ¯

2

(Y1·Y3) (Y2·Y4)

Nνµ+µ++µ1 128¯ ¯θµ

¯ ¯

2h

g2

L(Y1·Y3) (Y2·Y4)+g 2

R(Y1·Y4) (Y2·Y3) +gLgRa2m2µ(Y1·Y2)

i

Nνe/τ+µ++µ− 1 128|θe/τ|2

h f

gL2(Y1·Y3) (Y2·Y4)+g2R(Y1·Y4) (Y2·Y3) +gfLgRa

2m2

µ(Y1·Y2)

i

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tempera-A.2. MATRIX ELEMENTS FOR HNLS ABOVEΛQCD

A.2.2

Four-particle processes with leptons and quarks

Process (1+2→3+4) S SG−2

F a− 4|M|2

N+`+αU+D 1 128|θα|2|Vud|2(Y1·Y4) (Y2·Y3)

N+D`α+U 1 128|θα|2|Vud|2(Y1·Y2) (Y3·Y4)

N+U`α+D 1 128|θα|2|Vud|2(Y1·Y4) (Y3·Y2)

N+να→U+U 1 329 |θα|

2h16g2

R(Y1·Y4) (Y2·Y3) +(3−4gR)2(Y1·Y3) (Y2·Y4) +4gRθW(4gR3)a2m2

U(Y1·Y2)

i

N+U→να+U 1 329 |θα|

2h16g2

R(Y1·Y4) (Y2·Y3) +(3−4gR)2(Y1·Y2) (Y3·Y4) −4gR(4gR3)a2m2

U(Y1·Y3)

i

N+Uνα+U 1 329 |θα|2

h

16g2R(Y1·Y2) (Y3·Y4) +(3−4gR)2(Y1·Y4) (Y3·Y2) −4gR(4gR−3)a2m2U(Y1·Y3)

i

N+ναD+D 1 329 |θα|2

h

4g2R(Y1·Y4) (Y2·Y3) +(3−2gR)2(Y1·Y3) (Y2·Y4) +2gR(2gR−3)a2m2D(Y1·Y2)

i

N+Dνα+D 1 329 |θα|2

h

4g2R(Y1·Y4) (Y2·Y3) +(3−2gR)2(Y1·Y2) (Y3·Y4) −2gR(2gR−3)a2m2D(Y1·Y3)i

N+Dνα+D 1 329 |θα|2

h

4g2

R(Y1·Y2) (Y3·Y4) +(3−2gR)2(Y1·Y4) (Y3·Y2) −2gR(2gR3)a2m2D(Y1·Y3)i

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APPENDIX A. RELEVANT MATRIX ELEMENTS

Process (1→2+3+4) S SGF−2a−4|M|2

N→`α+U+D 1 128|θα|2|Vud|2(Y1·Y4) (Y2·Y3)

Nνα+U+U 1 329 |θα|2

h

16g2R(Y1·Y4) (Y2·Y3) +(3−4gR)2(Y1·Y3) (Y2·Y4) +4gR(4gR−3)a2m2U(Y1·Y2)

i

Nνα+D+D 1 329 |θα|2

h

4g2R(Y1·Y4) (Y2·Y3) +(3−2gR)2(Y1·Y3) (Y2·Y4) +2gR(2gR3)a2m2D(Y1·Y2)i

A.3

Matrix elements for HNLs below

Λ

QCD

In addition to interactions with leptons, HNLs will also decay into mesons.

A.3.1

Three-particle processes with single mesons

Process (1→2+3 or 1+2→3) S SGF−2M−N4|M|2

Nνα+π0 1 |θα|2fπ2

·

1− m 2 π M2N

¸

N`α+π± 1 2|θα|2|Vud|2fπ2

·µ

1−m 2 M2N

¶2

− m 2 π M2N

µ

1+m 2 M2N

¶¸

Nνα+η 1 |θα|2fη2

·

1− m 2 η M2N

¸

Nνα+ρ0 1 |θα|2¡1−2 sin2θW

¢2

fρ2

·

1+2m 2 ρ M2N

¸ ·

1− m 2 ρ M2N

¸

N→`α+ρ± 1 2|θα|2|Vud|2fρ2

·µ

1−m 2 M2N

¶2

+ m 2 ρ M2N

µ

1+m 2 M2N

−2m 4 ρ M4

N

¸

Nνα+ω 1 |θα|2¡43sin 2θ

W

¢2

fω2

·

1+2m2ω M2

N

¸ ·

1− m 2 ω M2

N

¸

Nνα+η0 1 |θα|2fη20

·

1−m 2

η0

M2N

¸

Nνα+φ 1 |θα|2¡43sin 2θ

W−1

¢2

fφ2

·

1+2m 2 φ M2N

¸ ·

1− m 2 φ M2N

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A

P

P

E

N

D

I

X

B

C

OLLISION

I

NTEGRALS

Consider the Boltzmann equation in comoving coordinates:

df1 dt =

df1 d lna

d lna

dt =

df1 d lnaH=

X

reactions

Icoll, (B.1)

with

Icoll=a

7−2Q 2gEf1

X

in,out

Z Ã Q Y

i=2

d3yi

(2π)32

f

Ei

!

S|M|2F[f](2π)4δ4(YinYout) (B.2)

The delta function can be rewritten as

δ4(Y

in−Yout)=δ4(s1Y1+s2Y2+...+sQYQ), (B.3) with si={−1, 1} if particle i is on the {left, right}-hand side of the reaction. The

Yi=aPi here are the comoving four-momenta.

B.1

Three-particle collision integral

Icoll= a

2Ef1

Z d3y

2d3y3 (2gπ)62

f

E22Ef3

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APPENDIX B. COLLISION INTEGRALS

B.1.1

Case

y

1

6=

0

Since a homogeneous and isotropic universe is assumed, only absolute values of momenta are relevant. Moreover, the matrix element in three particle interactions is independent of the four-momenta.

Icoll= S|M|

2a 8(2π)2g

f

E1

Z dy

2dy3dΩ2dΩ3y22y32

f

E2Ef3

F[f]δ4(s1Y1+s2Y2+s3Y3) (B.5)

Using the identity

δ3

(s1y1+s2y2+s3y3)= 1 (2π)3

Z

dλdΩλλ2ei(s1y1+s2y2+s3y3λ (B.6)

gives

Icoll= S|M| 2a 8(2π)5g

f

E1

Z dy

2dy3y22y32

f

E2Ef3

F[f]δ(s1Ef1+s2Ef2+s3Ef3

·

Z

dλλ2

Z

dΩλeis1y1λcosθλ

Z

dΩ2eis1y2λcosθ2

Z

dΩ3eis1y3λcosθ3

= S|M| 2a 8(2π)5g

f

E1

Z dy

2dy3y22y32

f

E2Ef3

F[f]δ(s1Ef1+s2Ef2+s3Ef3

·

Z

dλλ2

µ

4πsin(y1λ)

y1λ

¶ µ

4πsin(y2λ)

y2λ

¶ µ

4πsin(y3λ)

y3λ

= S|M| 2a (2π)2g

f

E1y1

Z dy

2dy3y2y3

f

E2Ef3

F[f]δ(s1Ef1+s2Ef2+s3Ef3

·

Z dλ

λ sin(y1λ) sin(y2λ) sin(y3λ) (B.7)

Rewrite the delta function of energies as

Z dy

3y3

f

E3 δ(s1Ef1+s2Ef2+s3Ef3)=

Z

dy3 y3

f

E3 δ¡

y3y3∗¢

y∗ 3 f E∗ 3 θ¡¡

(s1Ef1+s2Ef2 ¢

)2−a2m23¢

=

Z

dy3

y3 f E3 f E∗ 3

y3δ(y3−y

3)θ

µ ³

f

E∗3

´2

−a2m23

, (B.8)

where³Ef∗

3

´2

=¡y∗

3

¢2

+Mx22m23=

¡

s1Ef1+s2Ef2 ¢2

and y∗

3=

q

(s1Ef1+s2Ef2)2−a2m2

3.

Figure

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