Exploring hidden sectors: a phenomenological study of axions … CRISTIAN DAVID RUIZ CARVAJAL INSTITUTO DE FÍSICA FACULTAD DE CIENCIAS EXACTAS Y NATURALES UNIVERSIDAD DE ANTIOQUIA

E X P L O R I N G H I D D E N S E C T O R S : A P H E N O M E N O L O G I C A L S T U D Y O FA X I O N S A N D H I D D E N P H O T O N S M O D E L S

Tesis presentada al curso de posgrado

en Física de la Universidad de Antioquia

como requisito parcial para la obtención del grado

de Doctor en Física

C R I S T I A N D AV I D R U I Z C A RVA JA L

I N S T I T U T O D E F Í S I C A

F A C U LTA D D E C I E N C I A S E X A C TA S Y N AT U R A L E S

U N I V E R S I D A D D E A N T I O Q U I A

M E D E L L Í N -C O L O M B I A

Date: 28.01.2019

Advisor: Prof. Dr. Enrico Nardi

Co-adviasor: Prof. Dr. Oscar Alberto Zapata Noreña

Cristian David Ruiz Carvajal: Exploring hidden sectors: a phenomenologicalstudy of axions and hidden photons models, © 2018

C O N T E N T S

1 I N T R O D U C T I O N 12 H I D D E N S E C T O R S 5

2.1 Pseudoscalar Portal 62.1.1 The strong CP problem 62.1.2 Peccei-Quinn mechanism 82.1.3 Axion-like particles 132.1.4 Axion cosmology 152.1.5 Experimental approaches to search new bosonic WISPs 16

2.2 Vector Portal 192.2.1 Dark Photons 192.2.2 Dark photons in a light DM model 222.2.3 Experimental searches for DP 24

3 L I N K I N G A X I O N L I K E D A R K M AT T E R T O N E U T R I N O M A S S E S 273.1 Framework 28

3.1.1 Lagrangian 293.1.2 ZN symmetry anomaly cancellation 333.1.3 ALPs and sterile neutrino dark matter 35

3.2 Models 363.2.1 (2,2) ISS case 363.2.2 (3,3) ISS case 413.2.3 (2,3) ISS case 42

4 R E S O N A N T P R O D U C T I O N O F D A R K P H O T O N S I N P O S I T R O N

B E A M -D U M P E X P E R I M E N T S 474.1 Introduction 47

4.1.1 Beryllium anomaly 474.1.2 DP constraints 48

4.2 The PADME experiment at LNF 494.2.1 Experimental setup 50

4.3 A′ production via resonant e+e− annihilation 514.3.1 Effects of target electrons velocities 56

4.4 Results and discussions 585 C O N C L U S I O N S 67A C O N D I T I O N S T O C A N C E L T H E G AU G E A N D G R AV I TAT I O N A L

A N O M A L I E S I N T H E ZN S Y M M E T R I E S 71B e+ e− → A′ → e+ e− C R O S S S E C T I O N 73

B I B L I O G R A P H Y 75

iii

L I S T O F F I G U R E S

Figure 1 Feynman graph leading to the chiral anomaly. 6Figure 2 Sketch of a light shining through a wall experi-

ment. 17Figure 3 ALPs parameter space. 18Figure 4 Dark photon decay branching ratios into SM par-

ticles. 20Figure 5 Dark photons parameter space (visible decay). 25Figure 6 ALPs parameter space (benchmark regions). 39Figure 7 WW approximation and exact production for A′ in

e−Z → e−Z +A′ 49Figure 8 A′ production modes in fixed target e−/e+ beam

experiments. 50Figure 9 Schematic of the PADME Experiment. 51Figure 10 Evolution of the positron energy distribution 54Figure 11 The number of DP decaying outside the dump 55Figure 12 Positron annihilation probability 57Figure 13 Limits on the DP kinetic mixing ε as a function of

the mass mA′ 60Figure 14 Exclusion limits on the DP parameter space de-

rived from the E137 63Figure 15 Exclusion limits in electron beam-dump experiments. 64Figure 16 Feynman diagrams to the process e+e− → A′ →

e+e− 74

iv

L I S T O F TA B L E S

Table 1 Assignment of charges for two of the continuoussymmetries present in the Lagrangian of Eq. (3.1).The charges a, b, c and d are given by a = qd/2,b = sd + c, c = 1− rd and d = (p− q/2)−1 fromL (lepton number) conservation in the model. 34

Table 2 (D, vσ) values that can provide the total (secondcolumn) and partial (57 % in the third column)DM relic density by the ALP. 37

Table 3 Discrete and continuous charge assignments of thefields in the different models. 40

Table 4 Main features of the models discussed in the text.The constant g ∈ [10−3, 2] in the mass term ofthe ALPs, and the η Yukawa of order 10−2 in theM23a(b) models have been considered. The sec-ond and third columns represent the ALPs parame-ters (mass and coupling to photons, respectively),while in the last column are given the ISS massscales. 45

Table 5 Beam parameters for the Frascati BTF. The secondcolumn corresponds to the number of positrons ontarget per year, while the last two columns providethe energy limits of the beam. 52

Table 6 Number of 17 MeV DP produced in the first radia-tion length of a tungsten target for 1018 positronson target, for three different values of ε. The sec-ond and third columns are for a beam energy tunedto the resonant value Eres = 282.3MeV, assumingrespectively electron at rest and with the velocitydistribution in Eq. (4.8). The last column, also in-cluding ve effects, is for a beam energy tuned toEb = E res + 2σb, where σb/Eb ∼ 1% is the en-ergy spread. 58

v

1I N T R O D U C T I O N

Dark matter (DM) [1–3] and neutrino masses [4–9] are two of the manyexperimental facts that remain unexplained within the current fundamen-tal physics framework known as the standard model of particle physics(SM), and call for physics beyond it.As it has been established in the last decades, there is strong astrophysicalevidence that most of the matter making up the Universe is not ordinarymatter. Only a small part of the Universe (∼ 4%.) is constituted by particlesdescribed by the SM. The remaining part is divided magenta dark energy(∼ 73%) and DM (∼ 23%). The nature of this last is so far unknown.One of the most popular hypothesis among physicists is that it consistsof a new kind of matter, electrically neutral and stable, which does notinteract (at all or very weakly) with ordinary matter and therefore doesnot produce electromagnetic radiation, i.e., it is dark. Many theoreticalmodels envisage various types of DM, among which of particular interestis the possibility that there is a specific type of particle. This can acts asa “portal” between the dark particles of the new type of matter and theordinary matter of the SM through a new very weak interaction.In a theoretical scenario of DM particles, a new sector could be constitutedfor possible DM candidates that can be classified into three broad classes(for alternatives DM candidates in other scenarios see for example Refs.[10–14]):

1. Weakly interacting massive particles (WIMPs). Typically with massesof several GeV.

2. Ultra-light sub-eV particles introduced as a solution to the strongC P problem (axion) and similar pseudo-Goldstone bosons (axion-like particles or ALPs).

3. Sub-GeV hidden sector particles, neutral under SM forces and feeblyinteracting with the SM through new “portal” forces.

So far, DM searches have been primarily focused on WIMPs. Well-motivatedcandidates which have been long hunted for, and which remain of indis-putable interest [15–33]. However, due to the unsuccessful results we canconsider that there are at least two possible reasons why a tentative DMparticle of a new sector has not been discovered yet: first, the mass scale ofthe new particles, including the mediators of the new forces, is well abovethe energy scale reached so far in laboratory experiments, mainly investi-gated in collider experiments. Second, the mass scale is within experimen-tal reaches, but the couplings between the new particles and the SM onesare so feeble that the whole new sector has so far remained hidden. This

1

2 I N T R O D U C T I O N

last possibility often referred to as the “Hidden Sector” (HS) hypothesis,has triggered in recent years an increasing interest in many novel ideas tohunt for new physics at the intensity frontier [34–38] and corresponds tothe framework which is developed in this thesis. That is, a new scenariobeyond the SM (BSM) containing new particles with masses in the rangeof sub-eV until sub-GeV, neutral under the SM forces and interacting feeblywith the ordinary matter.Considering hidden sector physics, this thesis focuses on two main themes.In the first theme, we will analyze how the apparent absence of C P viola-tion in the QCD sector (strong C P problem) can be considered as a solidmotivation for going BSM. This problem, dynamically solved by the Peccei-Quinn mechanism, extends the SM gauge group with a global symmetry,and as a consequence of its breaking, a pseudo-Nambu-Goldstone boson,the axion arises [39–42]. In a similar idea, but without trying to solvethe strong C P problem, ALPs are also considered. In this case, ALPs arisefrom the spontaneous breaking of approximate global symmetries such,e.g. leptonic or family number symmetry, extensions of these or a mixtureof several additional symmetries [43, 44]. In one of our works, we usedthis concept to link these particles to neutrino masses through the mini-mal inverse seesaw (ISS) mechanism [45, 46] in order to explain the DMpuzzle. Specifically, we explore three minimal ISS cases [47, 48] wheremass scales are generated through higher dimension operators involvinga scalar field hosting ALPs. Remarkably, in one of the ISS cases, the DMcan be made of ALPs and sterile neutrinos, i.e. a possible two componentDM model [49].The second theme, inside the same concept of HS, is motivated from someexperimental anomalies like the discrepancy between the measured valueof (g −2)µ and the theoretical prediction [50] and the anomaly observedin decays of excited 8Be nuclei [51–53]. These puzzles can be explainedvia weakly coupled HS mediators with masses in the MeV range [54, 55].In particular, the last anomaly can be interpreted as due to the emission ofa 17 MeV HS boson, a dark photon (DP) A′, in the decay 8Be∗ →8Be+A′

followed by A′ → e+e−. This idea is developed in a second work, wherethe DP is considered as a natural candidate for super-weakly coupled newstate, since its dominant interaction with the SM sector might arise exclu-sively from a mixed kinetic term coupling the U(1)′ and QED field strengthtensors [56, 57]. In this work, we studied how DP’s can be searched whenits production is through the annihilations of e+ and e−, which can anni-hilate resonantly into an on-shell DP, or non-resonant into photon pairs,one of which ordinary and one dark (A′). The analysis is considered forthe PADME experiment, a new experiment under commission at the Fras-cati National Laboratories, aiming to seek this type of event through theaccurate reconstruction of the invariant mass distributions of e+e− pairsproduced in a possible decay of A′, or through the reconstruction of themissing mass in the balance between the initial state, constituted by thee+e− pair (using the positrons of the Frascati Beam Test Facility) and the

I N T R O D U C T I O N 3

final state in which only the ordinary photon is detected [58, 59]. A possi-ble confirmation for the DP interpretation of the 8Be anomaly, studyingthe inverse process e+e− → A′ at accelerators at the intensity frontier,would represent the discovery of HS and a particle physics breakthrough.Although, only recently the physics of HS has attracted a widespread in-terest from particle physicists, an international community of researcherspursuing these new directions is by now well established and is growingrapidly. Several experimental and theoretical proposals are presently be-ing put forth worldwide, and this clearly indicates that in the next yearsthe scientific interest in this direction will keep growing.

The rest of this thesis is present in four chapters. In chapter 2 some gen-eral aspects involved in an HS are discussed. Specifically, “pseudoscalarportal” is discussed in connection with the strong C P problem and thePeccei-Quinn symmetry as a possible solution. Besides the axion proper-ties, a generalization to ALPs is also commented, which just as the axion,can constitute the DM of our Universe. The same chapter also considersthe “vector portal” where a new massive vector boson, the DP, can mix withthe ordinary photon via kinetic-mixing and allows small coupling to electri-cally charged matter. In chapter 3, a framework linking axion-like particlesto neutrino masses through the minimal inverse seesaw mechanism is im-plemented in order to explain the DM puzzle. This chapter is based on thework “Linking axionlike dark matter to neutrino masses” [49]. In chapter4, the “vector portal” is invoked with the aim to explain through a DP the8Be anomaly. This chapter is based on the work “Resonant production ofdark photons in positron beam dump experiments” [59]. This work pro-vides a new DP production mode and was the fundamental idea for thepapers in refs. [60, 61], which are also discussed in this chapter. Finally,the conclusions of this thesis are presented in chapter 5.

This thesis has been based on the following works:[49] Linking axionlike dark matter to neutrino masses. Phys. Rev. D 96,115035 (2017)[59] Resonant production of dark photons in positron beam dump experi-ments. Phys. Rev. D 97, 095004 (2018)

2H I D D E N S E C T O R S

The SM provides a consistent description of all fundamental particles andtheir interactions (ignoring gravity). Nevertheless, it is also known to beincomplete, in particular, because new physics BSM must be responsible,among others, for issues as the neutrino masses and the DM. Some pos-sible explanations for these issues could be within an unknown HS (alsocalled dark sector). This can be defined as a collection of particles thatare not charged directly under the SM strong, weak, or electromagneticforces. Such particles are assumed to possess gravitational interactions,and may also interact with familiar matter through several “portal” inter-actions [34–36]. Their existence can be motivated by several theoreticaland observational puzzles, many of which are central in our quest to ob-tain a comprehensive understanding of the constituents of the Universeand their interactions. These include, among others, the nature of DM, thestrong CP problem, and a variety of astrophysical puzzles and DM-relatedanomalies.Although the definition of a HS is extremely broad, its physics can be ex-plored effectively and systematically by using some specific “portal” inter-actions constrained by the gauge and Lorentz symmetries of the SM. This“portals” are characterized by one or more mediator particles that makethe connection between the SM and the HS. The dominant SM-HS inter-actions depend on the mediator spin and parity and can be determinedby:

1. A pseudoscalar a(x), which can be identified as the axion or its gen-eralization to axion-like particle. This characterize the “pseudoscalar

portal” by operators like afa

FµνeFµν, a

faGbµνeGbµν,

∂µafaψγµγ5ψ (see sec-

tion 2.1).

2. A vector field A′(x), establishing the “vector portal” through the op-erator ε2 FµνF ′µν (see section 2.2).

3. A scalar S(x), establishing the “Higgs portal” through the operator(µS +λS2)H†H, where H is the SM Higgs doublet.

4. A fermion N(x), which gives the “neutrino portal” through the op-erator yN LHN , where N plays the role of a right-handed neutrino.

The operators in the last three items are renormalizable (dimension 4),while the ones in the first item are of dimension 5 and suppressed by somehigh energy scale fa.1 The “Higgs and neutrino portal” are best explored at

1 At the non-renormalizable level, additional portals can arise from dimension 6 operatorsinvolving a light (or even massless) vector mediator and SM fermions [62].

5

6 H I D D E N S E C T O R S

Jµ5 ∝ γµγ5

γα

γβ Gb

Ga

FIGURE 1: Feynman graph leading to the chiral anomaly. The blub on the leftvetex represents an axial vector coupling.

high-energy colliders and neutrino factories, respectively. While the “pseu-doscalar and vector portals” are principally explored at the intensity fron-tier (these can also be explored at the cosmic and energy frontiers). Inthe next sections we will consider these last two portals, even though inchapter 3 will be considered the “neutrino portal” to link this with ALPsthrough the minimal inverse seesaw mechanism in order to explain theDM puzzle.

2.1 P S E U D O S C A L A R P O RTA L

In this section, we will study the more important phenomenological prop-erties of the “pseudoscalar portal” considering the axion as a particularcase. This can be generalized to other related states, known as axion-likeparticles invoked in some scenarios where the spontaneous breaking ofapproximate global symmetries is considered.

2.1.1 The strong CP problem

The experimentally observed absence of the effect that characterizes CP-violation in the strong interactions, the neutron electric dipole moment,has been considered as a great puzzle of particle physics. The most ele-gant solution to this problem, proposed as a generic extension of the SMby Peccei and Quinn (PQ) [39, 40], leads to the prediction of a light pseu-doscalar particle: the axion. In the subsequent lines we will describe, ingeneral terms, the strong CP problem and the possible solution throughthe PQ symmetry. For a more detailed analysis, we refer to ref. [63].

In the limit of massless quarks, the strong interactions are invariant underan U(1)A axial transformation, q f → e

12 iαγ5q f , where f stands for the

quark flavor. The axial current associated to this symmetry is anomalousand gets quantum corrections from the triangle graph shown in Fig. 1.These corrections can be written as a total divergence that affect the QCDaction as:

δS = α

∫

d4 x∂µJµ5 = αg2

s N ′

32π2

∫

d4 x GbµνeGb,µν

︸︷︷︸

∂µKµ

, (2.1)

2.1 P S E U D O S C A L A R P O RTA L 7

where Kµ = εµαβγAbα

Gbβγ− gs

3 fbcdAcβ

Adγ

, N ′ is the number of mass-

less quarks, gs the strong interaction coupling, Gb,µν the gluon strengthfield tensor and eGb,µν ≡ 1

2εµναβGb

αβits dual field [63, 64]. Because of Eq.

(2.1) represents a surface integral, it is expected that (using appropriateboundary conditions)

∫

d4 x∂µKµ =∫

dσµKµ = 0, and the U(1)A axialsymmetry is recovered. Nevertheless, due to instantons effects, the vac-uum structure of QCD is non-trivial, and there are gauge configurationsin which

∫

dσµKµ 6= 0. Therefore, the U(1)A is not a symmetry of QCD[65, 66]. As a consequence of the non-triviality in the vacuum structure,the true minimum is a superposition of distinct vacuum configurations dis-tinguished by the so-called winding number. Then, due to vacuum to vac-uum transitions through instantons the following additional term in theQCD Lagrangian is generated [67, 68]:

LQCD ⊃ Θg2

s

32π2GbµνeGb,µν, (2.2)

where Θ ≡ Θ + arg det Mq, is a free parameter that characterizes theQCD ground state. In the above equation for Θ, the contribution givenby arg det Mq, in general of O(1), arises when the electroweak (EW) in-teractions are included.2 The term in Eq. (2.2) represent a CP violatingterm in the strong sector which leads to a neutron electric dipole mo-ment of order |dn| ∼ Θ 2.0 × 10−16 e cm [69, 70]. The experimentallimit, |dn| < 3.0× 10−26 e cm [71, 72], indicates that |Θ| ≤ 10−10, i.e.CP-violating effects in QCD are extremely small. This result confronts thenaive expectation that a dimensionless free parameter of the theory shouldbe O(1), and establishes the so-called strong CP problem: why is Θ sosmall ?There is no natural explanation for the extreme smallness of the parameterΘ, and sometimes this is treated as a fine-tuning problem. Some possiblesolutions have been contemplated to solve this problem:

1. The simplest possibility is considering a massless up-quark (nowruled out) [73, 74].

2. The so-called Nelson-Barr type of models [75, 76] that either requirea high degree of fine tuning, often comparable to setting |Θ| ≤ 10−10

by hand, or rather elaborate theoretical structures [77].

3. The PQ solution devised by Peccei and Quinn [39, 40] in which theCP-violating term vanishes dynamically.

With the aim of exploring the phenomenological aspects of axions as aparticular scenario in a possible “pseudoscalar portal”, we will considerthe PQ solution to the strong CP problem.

2 In the SM, the masses of the quarks arises from their interaction with the scalar Higgs. Thisinteraction is characterized by a complex matrix of Yukawa couplings so that the quarkmass matrix, Mq, is generally complex. By suitable transformations of the quark field, thedeterminant of Mq can be made real. This procedure, however, involves a global chiralphase transformation which leads to a term in the QCD Lagrangian similar to Eq. (2.2).


2.1.2 Peccei-Quinn mechanism

In the PQ mechanism, the Θ parameter is reinterpreted as a physical field:the axion a(x). In this scheme, one additional global chiral symmetryknown as the PQ symmetry, U(1)PQ, is introduced into the SM. This sym-metry is broken due to the axion’s anomalous triangle coupling to gluons,

LQCD ⊃

Θ−afa

g2s

32π2GbµνeGb,µν, (2.3)

where a(x) is the Nambu-Goldstone boson from the broken symmetry and

a

G

G

gagg

The aG eG couplingis the generic

feature of axions asopposed to other

light pseudoscalarparticles such ase.g., majoron or

axion-like particles.

fa = vPQ/N is the energy scale where the symmetry is broken, also knownas the axion decay constant, with N an integer that characterizes the coloranomaly [78–81]. In the original version, fa was postulated to be of the or-der of the EW scale implying very heavy axions, which were quickly ruledout by accelerator- and reactor-based experiments [82, 83]. Nevertheless,other models where fa is much higher than the EW scale, known as “invis-ible axion models”, are still viable (see section 2.1.2.1).Under the global shift, a → a + a0, in Eq. (2.3) is possible to absorb theΘ parameter in the definition of the axion field by the choice a0 = Θ fa,leading to a complete axion Lagrangian,

La =12(∂µa)2−

g2s

32π2 faaGb

µνeGb,µν, (2.4)

where we have included also a kinetic term for the axion field.The topological charge density ∼ ⟨trGµν eGµν⟩ 6= 0, induced by topologicalfluctuations of the gluon fields, such as QCD instantons, provides a non-trivial potential for the axion, which is minimised at zero expectation value,⟨a⟩ = 0, wiping out the strong CP violation. This dynamical realizationof CP-conservation in strong interactions is the main feature of the PQmechanism and leads to the above axion Lagrangian.In the PQ mechanism, although constructed as massless particles, the ax-ions do not remain massless in an effective low-energy theory. In fact, theaxion-gluon interaction in Eq. (2.4) induced by non-perturbative QCD ef-fects (such as QCD instantons) produce a mass for the axion approximatelygiven by:

ma∼=

mπ fπfa

pz

1+ z, (2.5)

where mπ = 135 MeV is the pion mass, fπ ≈ 92 MeV its decay constantand z =

mumd≈ 0.56 is the ratio between the up- and down-quark masses

[81]. The axion mass, besides of being suppressed by fa, must be protectedfrom quantum corrections by some symmetry (see e.g., section 3.1.2) sincethese breaks the underlying shift symmetry. Moreover, the axion interac-tions with the SM fields (suppressed by powers of fa), render the axiona light, weakly interacting and long-lived particle and therefore a natural


DM candidate [80, 84, 85]. The presence of the axion mass term impliesthat the field is promoted to be a pseudo-Nambu-Goldstone boson. Thisalso means that the Lagrangian in Eq. (2.4), at low energies, contains apotential, V (a), that to lowest order expands as 1

2 m2aa2. This potential

must respect the residual discrete shift symmetry a→ a+2nπ fa, for someinteger n, which remains because the axion is still the angular degree offreedom of a complex scalar field, parameterized as:

φ(x) =1p

2[ fa +ρ(x)]eia(x)/ fa , (2.6)

where ρ(x) is the radial component that gains a mass of O( fa). Assumingthat fa is greater than the EW scale, ρ(x) can be integrated out in aneffective low-energy theory [78–80].The result of a lowest order instanton computation gives for the axionpotential [86]: 3

V (a) = Λ4a

h

1− cos a

fa

i

, (2.7)

where Λa is the scale of QCD non-perturbative physics. Taking only smalldisplacements from the potential minimum, V (a) can be expanded as aTaylor series:

V (a) ≈Λ4

a

f 2a

a2 =12

m2aa2, (2.8)

where the leading order determines the mass term for the axion. Sincefa Λa, the axion mass is parametrically small. Some possible values forthese scales are contemplated in different scenarios. For example, in thecanonical QCD axion models, Λ4

a = Λ3QCDmu, with ΛQCD ≈ 200 MeV, mu

the u-quark mass, and fa is allowed to be:4

109 GeV ® fa ® 1017 GeV, (2.9)

where the lower and upper limit comes from supernova cooling [88–90]and black hole superradiance [91–93], respectively. Supernova limits canbe used to constrain axions by requiring that the axions produced do notcarry away energy equal to the total energy in the neutrinos emitted bythe supernova. The dominant production of axions comes from nuclearBremsstrahlung n + n → n + n + a, where the axion (a) is radiated bya neutron (n) and the neutrons scatter via a pion exchange. Additionally,cooling of stars can also be used to impose constraints on whether theaxions escape, or even just transport energy from the core to the outside of

3 A more accurate computation based on chiral perturbation theory (see Ref. [86]) gives

V (a) ∼s

1− F(mu, md ) sin2

a2 fa

, where F(mu, md ) is a (known) rational function of

the light quark masses.4 In others approaches, as e.g. string theory, fa typically takes values near the GUT scale,∼ 1016 GeV, though lower values of fa ∼ 1010−12 GeV are also possible [80, 85, 87].


the star. In this case, axions can be produced from Bremsstrahlung e+N →e + N + a, in which the axion is radiated from the electron (e) and N isa nucleus, or through the Primakoff radiation γ+ e → e + a, where thephoton (γ) is converted into an axion via the axion-photon coupling. Onthe other hand, the constraints coming from black hole superradiance arebased in a similar effect present in the Penrose process,5 where energy istransferred from a black hole into an axion cloud surrounding it (for moredetails see Refs. [91–93]).

2.1.2.1 Axion models

In this section, we will present “the classical” QCD axion models:

• The PQWW model, which introduces one additional complex scalarfield to the EW Higgs sector. It is known as the “visible axion model”(excluded experimentally).

• The KSVZ model, which introduces heavy quarks as well as a com-plex scalar singlet.

• The DFSZ model, which introduces an additional Higgs field as wellas a complex scalar singlet.

The last two models are called “invisible axion models” because the scaleof energy fa is much higher than the EW scale. These models are stillexperimentally viable.

PQWW model

The Peccei-Quinn-Weinberg-Wilczek (PQWW) model represents the origi-Having two Higgsdoublets, one gives

mass to the u-quark,while the other one

gives mass to thed-quark (one ofthem, or a third

doublet, givesmasses to the

leptons ).

nal version of the PQ mechanism [39–42]. Here, a doublet complex scalarfield, Φ(x), is added to the SM. The most general Lagrangian, invariantunder the U(1)PQ symmetry, allows that Φ(x) has a coupling with the SMparticles via Yukawa interactions. Just like the SM scalar field, Φ(x) hasa symmetry breaking potential and takes the vacuum expectation value(vev) determined by fa, ⟨Φ⟩ = fap

2, at the EW phase transition fa ∼ 246

GeV, implying heavy axions and large couplings. In this model, after thespontaneous symmetry breaking there are four real electromagneticallyneutral scalars: one gives the Z-boson mass, one is the SM Higgs, oneis the heavy radial Φ(x) field, and one is the angular Φ(x) field. This lastfield is the Goldstone boson of the spontaneously broken U(1)PQ symmetryand is identified as the axion. This has a coupling to the SM via the chi-ral rotations and the PQ charges of the SM fermions. The chiral anomalythen induces couplings to gauge bosons via fermion loops ∼ aG eG/ fa and

5 The Penrose process is a process by which an object is thrown into the ergosphere (a regionof space near to the horizon). Inside the ergosphere, it imparts negative energy to someparticle or part of the object, which is then thrown into the black hole while the rest ofthe object escapes. The object leaves with more energy than it had when coming in, whichextracts energy from the black hole.


∼ aF eF/ fa, where F is the EM field strength. The gluon term is the desiredterm and leads to the PQ solution of the strong C P problem. Notice thatall axion couplings come suppressed by the scale fa, which in this modelis fixed to be the EW vev. Since fa is too small, the axion couplings aretoo large, and it is excluded experimentally by, e.g., Rare meson decays[74, 82, 83, 94–96]. Since the QCD axions mixes with some of the neutralpions, it can appear in some of the rare meson decays. Even if it does notmix with the pions, as in the case of ALPs, it can still appear in decaysthrough its coupling to gauge bosons [97].

KSVZ model

Proposed by Kim [98] and independently by Shifman, Vainshtein and Za-kharov [99], include in the SM, besides a singlet complex scalar fieldφ(x), a color triplet, but SU(2)L singlet vector-like fermion Q. The moregeneral Yukawa interactions involving the PQ charge fields is given byλφQLQR + h.c., which provides the heavy quark mass. Because φ takesa vev of O( fa) the Q fields obtain a large mass, mQ ∼ λ fa, and there-fore can be integrated out. In this model, the SM particles are assumedto be uncharged under the U(1)PQ symmetry, while the singlet scalar fieldand the new color triplet are supposed to be charged. The U(1)PQ acts asa chiral rotation shifting the axion field, a, parametrized in the phase ofφ(x), Eq. (2.6), and corresponding to the Nambu-Goldstone boson fromthe breaking of U(1)PQ.At the classical level, the Yukawa interactions are unaffected by chiral ro-tations, and a is not coupled to the SM. However, at the quantum level,besides the axion-gluon interaction in Eq. (2.4), the axion field also has aphoton interaction given by:

Lint ⊃gaγγ

4FµνeF

µνa, (2.10)

where Fµν is the electromagnetic field-strength tensor and eFµν its dualtensor [81]. The coupling constant gaγγ is given by: In Eq. (2.11), Eq.(2.5)was used to get an expression that links the axion mass and the couplinggaγγ.

6

gaγγ =α

2π fa

EN −

23

4+ z1+ z

=α

2π

EN −

23

4+ z1+ z

1+ zp

zma

mπ fπ, (2.11)

with E and N the color and electromagnetic anomaly coefficients, respec-tively,

N =∑

i

X i T (Ri), E =∑

i

X iQ2i D(Ri), (2.12)

where T (Ri) is the Dynkin index of the SU(3)c representation of the fields,D(Ri) the dimension of the representation, and Q i , X i are, respectively,

6 In general, E/N is not known, so that for a fixed value of fa, a broad range of gaγγ valuesare possible [100, 101].


the electric and PQ charges of the fields. Since the number of color tripletsmust be equal to the number of antitriplets, N must be an integer (see ref.[81] for more details).The axion can also have an interaction with the fermions. This has a deriva-tive structure in order to remain invariant under the shift symmetry a →a+ a0. At low energies, it is possible to write the interaction as:

Lint ⊃gaN

2mNNγµγ5N∂µa+

gae

2meeγµγ5e∂µa (2.13)

where the first term represents the axion-nucleon interaction with a nu-cleon N (proton or neutron), while the last term describes the axion-electroninteractions [81, 102]. Notice here that the couplings to electrons, gae, andnucleons, gaN , are dimensionless and can be related to commonly-used di-mensionful couplings as: g ′ae,N = gae,N /(2me,N ) ∼ 1/ fa (for more detailssee ref. [102]).

DFSZ model

The Dine-Fischler-Srednicki-Zhitnitsky scheme [103, 104] is a hybrid be-tween the SM and the KSVZ model and couples the axion to the SM viathe Higgs sector. It requires, apart from the singlet complex scalar field,φ(x), at least two Higgs doublets, Hu and Hd , which give mass to theu- and d-quarks, respectively. Again, global U(1)PQ symmetry is imposedand spontaneously broken. Under this, only the new fields as well as theuR- and dR-quarks are supposed to be charged, allowing to build Yukawainteractions like:

Lint ⊃ Yi jqi LHd d jR + Γi jqi LeHuu jR + h.c.. (2.14)

in addition to the following potential:

V (φ) = −µ2φ |φ|

2 +λφ |φ|4 +λ3H†d Huφ

2. (2.15)

After the PQ and EW symmetry breaking, the parameters in the potentialmust be chosen such that the SM Higgs fields remain light, consistent withthe observed 125 GeV and the EW vev. Additionally, axial current cou-plings between the axion and SM fermions are generated, inducing alsothe coupling to G eG via the colour anomaly. In this case, the same fermionloops induce the axion-photon coupling, aF eF , which is computed via theelectromagnetic anomaly and corresponds to the sum of quark and leptonFreedom in this

model appearsthrough the lepton

charges: we are freeto choose whetherit is Hu or Hd thatgives mass to the

electron viaHu,d l LeR.

loops (different lepton PQ charges give different values for the anomaly,and thus the coupling) [81].The doublet and singlet scalar fields can be parametrized as:

H0d(x) =

vd + hd(x)p

2exp

h

iζ(x)

v+ Xd

a(x)fa

i

,

H0u(x) =

vu + hu(x)p

2exp

h

iζ(x)

v− Xu

a(x)fa

i

,

φ(x) =fa +ρ(x)p

2eia(x)/ fa , (2.16)


where the phases of the fields involve the Nambu-Goldstone boson ζ(x)eaten by the Z0 to generate its mass, and the Nambu-Goldstone bosona(x) identified as the axion field.

An important difference between KSVZ and DFSZ models is that, for DFSZthe term ∼ G eG is induced by light quark loops calculated at low energy,rather than via the heavy quarks.

2.1.3 Axion-like particles

The proposal of an anomalous U(1)PQ symmetry is motivated to provide asolution to the strong CP problem. This concept has been generalized toother similar weakly interacting slim (or sub-eV) particles (WISPs), as e.g.,Axion-Like Particles (ALPs). However, this scenario do not try to solve thestrong CP problem. ALPs can arise as pseudo-Nambu-Goldstone bosonsfrom the breaking of other global symmetries such, as family symmetries.In this section, some relevant features of ALPs are discussed.

Theoretically, ALPs arises as pseudo-Nambu-Goldstone bosons from thebreaking of some chiral global symmetry, U(1)X .7 In some models, U(1)X

is taken as the leptonic or family number symmetry, extensions of these,or a mixture of several additional symmetries.The ALPs-SM interactions are similar to the axion case in the sense thatthey are very weak. Their interactions are suppressed by the energy scalewhere the symmetry is broken (¦ 109 GeV) [113–115]. In particular, theinteraction with the photons, at low energies, is given by:

Lint ⊃gaγγ

4FµνeF

µνa, (2.17)

while the coupling to gluons is neglected since the ALPs do not try to solvethe strong CP problem. From Eq. (2.17), it is possible to see that, in fact, theaxions and ALPs have the same interaction with the photons, Eq. (2.10).However, in the axion case, the gaγγ coupling is proportional to the axionmass, Eq. (2.11), while, for the ALPs this relation is missing. In this case,the effective coupling is given by:

gaγγ =α

2πvσCaγγ, (2.18)

where α is the fine-structure constant and vσ the energy scale where U(1)X

is broken, which can take values in a similar range to those given in Eq.(2.9). Caγγ is the anomaly coefficient in the U(1)X chiral symmetry andreads:

Caγγ = 2∑

ψ

XψL− XψR

C (ψ)em

2, (2.19)

7 This symmetry is considered approximate and accidental in the sense that it is broken byoperators of high dimension [105–112].


where XψR,Lare the field charges under the chiral symmetry and C (ψ)

em theelectric charge [81]. This anomaly coefficient is in general of order oneand is used to determine the width of the region where ALPs can be consid-ered as DM candidates (see discussions in chapter 3). As a consequence ofthe ALPs-photon coupling, ALPs and photons can oscillate into each other[116–118]. The interaction in Eq. (2.17) can be written as:

Lint ⊃ gaγγE ·B a, (2.20)

where B plays the role of an external magnetic field, while E describes thephoton field that couples to ALPs. From Eq. (2.20), considering E perpen-dicular to the momentum of the photon, only the transverse component ofB couples to photon. Consequently, the mass matrix of the ALPs-photon sys-tem is not diagonal and therefore the propagation eigenstates are differentfrom the interaction eigenstates. This fact gives rise to ALPs ↔ photonsoscillations, with an oscillation probability Pa↔γ ∼ (gaγγBz)2, where B isMost of the

experimentaltechniques to

search ALPs/axionsare based on the

idea of oscillation(see section 2.1.5).

the intensity of the external magnetic field which is extended over a dis-tance z [119]. Since the axion and ALPs share the same interaction withthe photons, this oscillation phenomena is also valid for the axion field.The ALPs can be also parametrized as the phase of a singlet scalar field, Eq.(2.6). With this parametrization, a mass value for the ALP can be generatedthrough a non-renormalizable operator of higher dimension D (D 4),suppressed by the Planck scale [120, 121], i.e.,

L ⊃ gσD

M D−4Pl

+ h.c., (2.21)

where g is a constant of order one, and MPl = 2.4× 1018 GeV is the re-duced Planck scale. After the symmetry breaking, replacing in Eq (2.6)φ → σ and fa → vσ, and substituting the expression for σ in Eq.(2.21),the Lagrangian becomes:

L ⊃g

M D−4Pl

12D/2

(vσ+ρ)D

ei avσ

D + e−i avσ

D

=g

M D−4Pl

12D/2

(vσ+ρ)D2cos

avσ

D

=g

M D−4Pl

12D/2

vDσ2

1−12

D2

v2σ

a2 + ...

, (2.22)

where in the second equality the Taylor expansion for the cosine has beenused. From this expression we can identify an ALP mass term given by:

ma∼= |g|

12 D

vσp2

h vσp2MPl

iD2 −2

. (2.23)

Notice here that the ALP mass can be very small since in Eq. (2.23) a Planckscale suppression is considered.


2.1.4 Axion cosmology

Thermally producedWISPs are alsopossible, but theseare disfavored byobservations of theUniverse’slarge-scale structure[84, 122].

In the framework of a broad class of inflationary scenarios, axions andother WISPs like ALPs and hidden photons may be non-thermally producedin the early Universe though the vacuum-realignment mechanism and sur-vive as constituents of the dark Universe.8 In some models, these can arisealso via the decay of topological defects such as axion strings and domainwalls [80, 84, 85, 124]. In the subsequent section, we will describe thevacuum-realignment mechanism in the particular case of axions. However,this can be generalized to the other bosonic WISPs.

Vacuum-realignment mechanism

At early times, at temperatures well above the QCD phase transition, theaxion is effectively massless and the corresponding field can take any value,parameterized by the so-called misalignment angle θa. Later, as the temper-ature of the primordial plasma falls below the hadronic scale, T < ΛQCD,the axion develops its mass ma due to non-perturbative effects. When thismass becomes of order of the Hubble expansion rate, H, the axion fieldbegins to oscillate around its mean value ⟨a⟩= 0, with the axion mass de-termining the oscillation frequency. These coherent and spatially uniformoscillations correspond to a coherent state of nonrelativistic axions. The os-cillations can behave as a cold DM (CDM) fluid since their energy densityscales as ρa ∝ a−3, i.e. the same behavior as ordinary matter. The axioncontribution to relic density depends on the order in which cosmologicalevents take place, especially whether the breaking of the PQ symmetry oc-curred before or after inflation [80, 84, 85]. This implies that the densityin the misalignment population is fixed by the initial field displacement.Let’s consider both scenarios.

• Before inflation: Assuming that the reheating temperature after infla-tion is below fa and there is no dilution by, e.g. late decays of BSM(e.g. moduli), the expected energy density can be estimated to be: QCD axion with

fa ∼ 1016 GeVwould overclose theUniverse, unless theinitial misalignmentangle is very small,θa ∼ 10−3 [43, 84].

Ωah2 ≈ 0.741×

fa

1012 GeV

7/6θa

π

2

, (2.24)

where θa ∈ [−π,π], completely random, is the initial misalignmentangle. From this expression, for generic values of the misalignmentangle, θa ∼ O(1), the axion could be the main constituent of CDMin the Universe if its decay constant is fa ∼ 1011−12 GeV.

8 The misalignment mechanism can be considered in the case of hidden photons only insome specific cases. That is, for example, when they have an explicit mass term, not fromspontaneous symmetry breaking. In this case, the hidden photon is not a U(1) gauge field,although the model is still consistent. In the Stuckelberg model, for example, and undersome specific additional conditions, the hidden photons can also give rise to a Bose-Einsteincondensate from misalignment mechanism and contribute to (or account for the whole)DM (see refs. [84, 123] for more details).


• After inflation: In this framework, favorable for models with fa ®1014 GeV [84, 85], if the reheating temperature after inflation isabove fa, the initial misalignment angle takes different values in dif-ferent patches of the Universe, resulting in an average misalignmentangle θ2

a∼= π2/3. In this case, the energy density can be estimated

by:

Ωah2 ≈ 0.16

θa

π

2

×

ma

eV

1/2fa

1011 GeV

2

. (2.25)

Is important to comment that Eq.(2.24) and Eq.(2.25) are also valid ex-pressions in the case of ALPs as DM candidates. In particular, Eq. (2.25)will be used in chapter 3 to determine the relic density in terms of the ALPmass and the dimension D of the non-renormalizable operators consideredin Eq. (2.21).The regions in the parameter space (defined by gaγγ and ma) where theaxion (and ALPs) can be considered as CDM candidates are shown in Fig.3 (for a post-inflationary scenario), together with some excluded experi-mental regions that we will discuss in the following section.

2.1.5 Experimental approaches to search new bosonic WISPs

The new physics not necessary resides at high energies (above TeV). Itcould well be found at the “low-energy frontier”, and be accessible withintensity-frontier tools. Indeed, experiments that use intense beams of pho-tons, charged particles, and/or sensitive detectors may be used to produceand study new feebly-interacting particles that lie around or well below theweak scale.In addition, a set of

cosmologicalconstraints frommodification of

big-bangnucleosynthesis,

distortions of thecosmic microwave

background andextragalactic

background lightmeasurements

exclude a largeregion of the WISPs

parameter spaceand are sensitive to

very smallcouplings to

photons [125, 126].

The experimental setup considered to search new bosonic WISPs as ALPs,axion and hidden photons can be located primordially in the intensity fron-tier (low energy). However, high-energy colliders are also sensitive to alarge region in the bosonic WISPs parameter space [127–129]. In this case,the different production mechanisms at colliders offer a rich phenomenol-ogy allowing us to probe a large range of masses and couplings.9

Depending on the region in the WISPs parameter space, the search strate-gies vary greatly. For intermediate masses up to the GeV scale, colliderexperiments searching for missing-energy signals probe long-lived WISPswith non-negligible couplings to SM particles. Current and future beam-dump searches are sensitive to masses below ∼1 GeV radiated off photonsand decaying outside the target [136–139]. For masses below twice theelectron mass, the WISPs can only decay into photons and the correspond-ing decay rate scales like the third power of the mass. Thus, light WISPsare mostly long-lived and travel long distances before decaying. Usually, in

9 Besides resonant production, bosonic WISPs can be produced in decays of heavy SM par-ticles [128–131] or in association with gauge bosons, Higgs bosons or jets [132–135].


Laser Detector

Wall

−→B γa

γ

−→B

FIGURE 2: Sketch of a light shining through a wall experiment. In this sketch,the left and right boxes represent the production and regeneration regions,respectively. In these, a transverse magnetic field, necessary to the conversion,is identified with the vertical arrows. The central block, labeled as the wall,represents the optical barrier for the laser sent from the “laser” box. Finally, the“detector” box represent the place where can be detected the photons comingfrom the regeneration region as a consequence of the axion-photon conversion.

this case the experiments employ intense sources and ultra-sensitive toolssuch as lasers, microwave cavities, radio-frequency and strong electromag-netic fields to detect the particles. The fundamental key is the photon cou-pling gaγγ, in the case of ALPs/axions, or the kinetic mixing ε in the caseof hidden photons (see section 2.2). Since these couplings allow to haveWISPs↔ photons oscillations, it is possible to explore different regions inthe parameter space. At this point is important to say that the constraintsover the axion parameter space are stronger than the constraints associ-ated with ALPs, since for ALPs the mass and coupling are two independentparameters. Moreover, couplings to other SM particles are generally lessconstrained than the photon coupling [102, 140–144].The searches of a new bosonic WISP can be considered, among others,within experiments such as light shining through a wall, Haloscopes or He-lioscopes. The idea behind these experiments as, e.g. light shining through In the particula case

of QCD axion, itsmass is related tothe PQsymmetry-breakingscale fa viaEq.(2.5). Therefore,higher values to facorrespond tolighter QCD axionmasses, which arewell beyond thereach of traditionalexperiments likeADMX (wouldrequire much largercavities).

a wall (see Fig. 2), is based in a laser sent through strong magnetic fieldsallowing photon → WISPs oscillation (in the case of hidden photons themagnetic field is not necessary). Due to the very weak interaction of WISPswith the matter, these particles can pass through a wall, reconverted intophotons in the presence of another strong magnetic field, and finally, aredetected at the end of the way [147, 157–160]. In a similar procedure, con-sidering the WISPs as a possible DM particle coming from the Milky Wayhalo, the Haloscopes searches these particles making use of microwaveresonant cavities [161, 162].On the other hand, the use of astrophysical objects as a likely source ofWISPs is considered by the Helioscopes. This scheme admits the same ideaof oscillation discussed above. The difference here is that in this case, thewall is given by some object between the source and the earth (everythingbetween the astrophysical objects considered as the source of WISPs, i.e.,part of the astrophysical object, the atmosphere, and so on) [149, 163–166]. These and others experimental techniques (see ref. [38, 141]) giveconstraints on the WISPs parameter space. In particular, we show in Fig3 the ALPs parameter space. The figure shows excluded regions (blue re-


Axion

CDM

Massive Stars

SN1987A γ-Ray Burst

γ-Ray TransparencyIAXO

ALPS-II

Halo

scop

es

3.55keV

Linefrom

DecayingALPDM

ALPCDM

ABRA-Res.

ABRA-Broad.

-12 -10 -8 -6 -4 -2 0 2 4

-18

-16

-14

-12

-10

-8

Log10 ma [eV]

Lo

g10|g

aγ|[

Ge

V-

1]

FIGURE 3: ALPs parameter space. This figure shows some excluded regions fromthe non-observation of an anomalous energy loss of massive stars due to ALPs (oraxions) emission [145], of a γ-ray burst from SN 1987A due to conversion of anALPs in the galactic magnetic field [88, 89, 146] and of DM axions or ALPs con-verted into photons in microwave cavities placed in magnetic fields [147–150].It is also showed the red band where the ALPs may constitute all of CDM (ALPsCDM), and the regions where the ALPs may explain the cosmic γ-ray transparencyand the x-ray line at 3.55 keV [151–155]. The green regions are the projected sen-sitivities of the light shining through a wall experiment ALPS-II, of the helioscopeIAXO, of the haloscopes ADMX and ADMX-HF [147, 149]. The black (green) solidline in the lower left corner shows the sensitivity of the proposed ABRACADABRAexperiment [156] using a resonant (broadband) circuit. Furthermore, it is showedthe region of QCD axion (the yellow band) which was recently extended to coverthe lower right corner [100] studied in the context of some realistic axion models,with the region below the orange solid line corresponding to the axion CDM.

gion) from the non-observation of an anomalous energy loss of massivestars due to ALPs (or axions) emission [145], of a γ-ray burst from SN1987A due to conversion of an ALPs in the galactic magnetic field (purpleregion) [88, 89, 146] and of DM axions or ALPs converted into photonsin microwave cavities placed in magnetic fields (gray regions) [147–150].Additionally, it also showed the regions where the ALPs may explain thecosmic γ-ray transparency (region delimited by the red lines at the leftside) and the x-ray line at 3.55 keV (vertical red line at the right side)[151–155], as well as, the band where ALPs are considered as CDM can-didates (red band labeled as ALPs CDM). The green regions are the pro-jected sensitivities of the light shining through a wall experiment ALPS-II,of the helioscope IAXO, of the haloscopes ADMX and ADMX-HF [147, 149].Furthermore, it is showed the region of QCD axion (the yellow band),with the region below the orange solid line corresponding to the axion

2.2 V E C T O R P O RTA L 19

CDM. The black (green) solid line in the lower left corner shows the sen-sitivity of the proposed ABRACADABRA experiment, acronym for A Broad-band/Resonant Approach to Cosmic Axion Detection with an AmplifyingB-field Ring Apparatus [156]. This is an experiment that make use of aresonant (broadband) circuit to exploit the fact that axion DM, in the pres-ence of a static magnetic field, produces response electromagnetic fieldsthat oscillate at the axion Compton frequency. Since the traditional experi-mental setup of Haloscopes is only sensitive to axion DM whose Comptonwavelength is comparable to the size of the resonant cavity, this is a newproposal for axion DM detection in a mass range ma ∈ [10−14 − 10−6] eV.For more specific details about the experiment, we invite the reader to seeref. [156].

2.2 V E C T O R P O RTA L

In this section, we discuss the theory and physics motivation to study theparticle that establishes the “vector portal”: the dark photon. An overviewof the experimental searches for this particle is also considered.

2.2.1 Dark Photons

Dark photon, also known as hidden photon or A′ boson, is a massive vectorboson coming from a new U(1)′ gauge symmetry. This particle can mixwith the ordinary photon via a kinetic-mixing term [56, 57]. This mixingallows SM photon-A′ oscillations (in a similar way to the ones discussedin section 2.1.3 for ALPs) and produces a small coupling between A′ andthe electrically charged matter. This new gauge boson implies a new extraforce that we have not seen for multiple reasons. In principle, the DP canhide from observation in experiments because of two main reasons:

1. This new particle is very heavy (around TeV). Therefore, a lot ofenergy is necessary to create it. Moreover, forces mediated by heavyparticles are of extremely short range;

2. The interactions with the SM particles are extremely weak. In thiscase, their effects would simply be too feeble to have been observedso far.

According to that, the DP is often referred to as belonging to a hiddensector.DP’s are characterized by a kinetic mixing that produces an effective parity-conserving coupling of the DP to the electromagnetic current JµEM ,

L ⊃ εeA′µJµEM , (2.26)

suppressed relative to the electron charge e by the parameter ε, whichcan be naturally small. This parameter can take values in the range ∼


200 400 600 800 1000 1200 1400 160010-5

10-4

0.001

0.010

0.100

1

Dark Photon Mass (MeV)

BR

ηℽπ0ℽπ+π-π+π-π+π-π0π0π+π-π0ωπ0→2π0+ℽK0K0K+K-π+π-μμee

FIGURE 4: Dark photon decay branching ratios into SM particles. The DP maycouple also to other non-SM particles in the hidden sector. For example, there maybe new matter states charged under the new U(1)′ which may include particlesthat constitute DM. If decays of the DP to the hidden sector states are kinemat-ically forbidden such couplings are irrelevant to the phenomenology and thesebranching ratios are hold (figure taken from ref. [170]).

10−4−10−2(∼ 10−6−10−3), if the mixing is generated by one (two)-loopsinteraction [167–169]. The effective interaction in Eq.(2.26) comes fromconsidering an extra U(1)′ gauge group under which all the SM particlesare uncharged. Therefore, the dominant interaction with the ordinary mat-ter is only via the kinetic mixing with the hypercharge U(1)′ gauge boson.At low energy (below the electroweak scale), the mixing is directly withthe SM photon and the effective Lagrangian can be written as:

Le f f = LSM −14

F ′µνF ′µν+12

m2A′A′µA′µ−

ε

2F ′µνFµν, (2.27)

where F ′µν is the field strength of the hidden gauge field A′µ, Fµν the SMphoton field strength and mA′ the mass of the DP [56]. From this equationDP masses can also

be generated viathe Sütckelberg

mechanism, whichis especially

relevant in stringtheories [171].

we can see that the kinetic mixing and the mass of the DP are the onlynew two parameters. The mass can arise via the Higgs mechanism andcan take a large range of values (from sub-eV until GeV) [34, 172]. Anatural dividing line in the range of masses is given by mA′ ∼ 2me ∼ 1MeV [34–36].

• For mA′ > 1 MeV, a DP can decay to electrically charged particles,e.g., e+e−, µ+µ−, or π+π−, or to light hidden-sector particles (ifavailable), which can in turn decay to ordinary matter. Fig 4 showsthe predicted branching ratios of A′ into SM particles. This case isreferred to as the visible DP model. If the DP decay into hidden sectorstates, the model is referred to as the invisible decay model.

mA′ ∼MeV-GeV could explain some experimental anomalies like theanomaly observed in decays of excited 8Be nuclei [51–53] and thediscrepancy between the measured value of (g − 2)µ and the the-oretical prediction [50]. Such an A′ can be efficiently produced in


electron (positron) or proton fixed-target experiments, and at e+e−

and hadron colliders [34, 37].

• For mA′ < 1 MeV, the DP decay to e+e− is kinematically forbidden,and only a much slower decay to three photons is allowed. At verylow masses, the most prominent implication coming from the kineticmixing is that the propagation and the interaction eigenstates aremisaligned (similar to the case present in both neutrino and ALPsoscillations). This gives rise to γ↔ A′ oscillation phenomena thatcan be searched by light shining through a wall experiments and/orHelioscopes (in this case the use of magnetic fields to trigger theconversion is not required). The oscillation mechanism can also gen-erate the required A′ relic density for them to account for all theDM. This hypothesis can be tested in direct DM detection experi-ments or indirectly through the A′ decay into three photons whichcould be observed above the astrophysical diffuse X-ray backgrounds[173, 174].

With the aim of understanding the physical consequences of the kineticmixing term it is convenient to remove this by a suitable field re-definition.There are two simple ways to eliminate the kinetic mixing term [57]:

1. Making the field re-definition: Aµ→ Aµ− εA′µ;

2. Through the shift: A′µ→ A′µ− εAµ.

Although the resulting physics is, of course, completely equivalent, thepicture resulting from both shifts is somewhat different. Depending on thesituation it is often suitable to use one or the other picture.

Case 1

Inserting the above first shift in Eq. (2.27), the kinetic mixing term is re-moved and the Lagrangian becomes10

Le f f ⊃ −14

FµνFµν−14

F ′µνF ′µν+12

m2A′A′µA′µ+ eJµEM (Aµ−εA′µ). (2.28)

In this case, we have two independent particles. A massless particle (theSM photon A) and a massive uncharged vector particle, A′. Notice herethat from the last term in Eq. (2.28) the A′ has a coupling with the ordinarymatter via the electromagnetic current, Eq. (2.26), such that a particle withelectromagnetic charge Q now also carries a “hidden charge” Q′ = εQ,

10 Is also possible to have a mass mixing (parametrized by εZ ) between the new vector bosonA′ and the heavy Z boson of the SM. In this case, there is a coupling between the A′ and theweak neutral currents, JNC

µ , of the SM via −εZg

2 cosθWJNCµ A′µ. This additional interaction

violates parity. Consequently, new phenomena such as “dark parity violation” in atoms andpolarized electron scattering can occur [175, 176]. Enhancements in rare dark decays ofthe Higgs as well as K and B mesons into A′ particles can also occur, suggesting new exper-imental areas of discovery. However, since for small mA′ the parameter εZ is suppressedby (mA′/mZ )

2, these effects are extremely small and can generally be neglected.


coupling it to the hidden photon A′. As a consequence, the interactionbetween two charged particles, with charges q1e and q2e separated by adistance r, can be estimated by:

V (r) = q1q2α

r

h

1+ ε2e−mA′ ri

, (2.29)

where the first term is the ordinary Coulomb interaction, while the secondterm is the additional contribution from the “new force” mediated by thehidden photon A′ [57]. Note that, for r > 0, the “new force” becomesnegligible whenever the mass mA′ 1/r. In this case, the force has a rangeshorter than the resolution and is therefore not observed. Alternatively, ifε is very small the force becomes simply very weak and at some pointunobservable, even if mA′ is small and the range is large.Since the new massive vector particle A′ has a coupling to charged par-ticles according to Q′ = εQ, the A′ can be produced in scattering exper-iments with charged particles. These interactions are considered in thefixed-target experiments and will be discussed in section 2.2.3.

Case 2

On the other hand, considering the second way to remove the kinetic mix-ing term in Eq. (2.27), that is, inserting the shift A′µ → A′µ − εAµ, theLagrangian becomes to:

Le f f ⊃ −14

FµνFµν−14

F ′µνF ′µν+m2

A′

2(A′µA′µ−2εA′µAµ+ε2AµAµ)+ eJµEM Aµ.

(2.30)

Now, in contrast to the previous case 1, the charged particles do not havea “hidden charge”. However, a non-diagonal mass term arises mixing theA′ with the SM photon A. This mass term leads to γ ↔ A′ oscillationssimilarly to the non-diagonal mass matrices of neutrinos (also similar toALPs case discussed in section 2.1.3).

2.2.2 Dark photons in a light DM model

Physics BSM, emerging as a whole new sector and containing new parti-cles as well as new interactions, do not need to be at particularly high-energies. New particles masses could well be within experimental reach,provided these particles couple sufficiently feebly to the SM particles. Forexample, states with mass below 1 GeV would have easily escaped detec-tion by underground experiments seeking for halo DM, so that comple-mentary searches attempting to cover this mass region are well motivated.An interesting scenario considers light DM with mass in the ∼(1 MeV –1 GeV) range,11 charged under a new U(1)′ symmetry and that interacts

11 Couplings between DM and hidden vector bosons at the MeV-GeV scale can drasticallymodify the phenomenology of DM. In direct detection, the scattering cross section can be


with the SM particles via the exchange of a vector boson A′. Dependingon the relative mass of the A′ and the DM, the DP can decay only into SMparticles (visible decay) or dominantly to light DM states (invisible decay).In many models of DM, the DM particles do not have direct couplings to theSM. Instead, they interact with the SM through a mediator, a particle thatcouples to both the SM and the DM. As was discussed previously, the gaugeand Lorentz symmetries of the SM restrict the ways in which the mediatorcan couple to the SM. If the mediator is a vector force carrier (referredto as a DP) from an additional U(1)′ gauge group under which DM ischarged, a kinetic mixing interaction is gauge invariant under both U(1)′

and U(1)Y . Consequently, a mixing ε2 F ′µνFµν is induced and responsible for

the phenomenology considered in a light DM model. In this case, the DM The nature of theDM particle contentis important inestablishingconstraints on themodel, as well asthe parts ofparameter spacefavored bycosmological andastrophysicalobservations of DM.

can be a fermion, χ, or a scalar boson, Φ, that couples to the DP throughthe hidden sector gauge interactions:12

LDM f erm.= χ(i /D−mχ)χ, (2.31)

LDMscal.= (DµΦ)∗(DµΦ)−m2

Φ|Φ|2, (2.32)

where Dµ ≡ (∂µ − i g ′Aµ) and g ′ is the dark sector coupling between themediator and the light DM [34]. This model has a broad parameter spacewhere several specific regions are particularly important targets, either be-cause they are theoretically well-motivated, or because exploring theseregions decisively tests outstanding anomalies [34, 35, 37, 57].Thermal freeze-out of DM annihilations into ordinary matter sets the mainbasis for the “WIMP Miracle” in TeV-scale weakly-interacting DM models.However, it can be also applied to DM particles in a hidden sector scenario.For all types of mediators and light DM candidates (χ) there are two im-portant annihilation processes to consider. The secluded annihilation topairs of mediators via χχ → A′A′ (for mA′ < mχ), followed by mediatordecays to SM particles [177] and direct annihilation to SM final states viavirtual mediator exchange in the s-channel without an intermediate step,χχ → A′∗→ SM SM (for mχ < mA′). In the first case, the annihilation ratescale as ⟨σv⟩ ∼ g ′4/m2

χ and there is no dependence on the SM-mediatorcoupling εe. Since extremely small values of this coupling can be compat-ible with thermal light DM in this regime, the secluded scenario does notlend itself to decisive laboratory tests [34]. On the other hand, in direct an-nihilation the rate scales as ⟨σv⟩ ∼ [g ′2αε2m2

χ ]/(m4A′). This regime offers

a predictive target for discovery (or falsifiability) since the dark couplingg ′ and mass ratio mχ/mA′ are at most O(1). Therefore, there is a min-imum SM-mediator coupling (εe) compatible with a thermal history. Analternative to the thermal DM scenario is one where the DM abundance is

increased due to the light mediator or, alternatively, the kinematics of the scattering can bealtered. In indirect searches, the self-annihilation and self-scattering rates for the DM canboth be enhanced at low velocities; the former can lead to striking signals in cosmic rays,photons, and neutrinos, while the latter can significantly modify the internal structure ofDM halos [34, 35, 37, 57].

12 Majorana mass terms for χ may also be allowed. However, because these mass terms mustvanish with the restoration of the gauge symmetry they are naturally small.


established via freeze-in [178]. In this scenario, the dark sector is never inthermal equilibrium with the SM, but out-of-equilibrium scatterings grad-ually populate the DM. Because thermal equilibrium is not established be-tween the dark and visible sectors, the couplings in freeze-in scenarios aretypically very small. An usual example for this case is to consider a kineti-cally mixed DP, which can decay into two DM particles χ [61]. A variationof this model is to take mA′ to be tiny, mA′ eV. In this case, the DM abun-dance is also built up slowly over time through SM particles annihilatingto DM particles through an off-shell A′. However, the final DM abundanceis independent of mA′ , and there is only a mild dependence on mχ above∼1 MeV (see ref. [34] for more details).

2.2.3 Experimental searches for DP

Searches for these new particles are part of the intensity frontier since in-tense beams of electrons, positrons or protons are required to produce DPin sufficient quantities to compensate for their weak coupling to ordinarymatter. Several experimental proposals exist, including the experimentaltechniques discussed in section 2.1.5 (with some modifications e.g., ne-glect the magnetic field). Current and planned DP searches are character-ized by their strategies for production and detection. The main DP produc-tion channels can include:

• Process like-Bremsstrahlung, e±Z → e±Z +A′, for electrons (or positrons)incident on a nuclear target of charge Z [179];

• Resonant e+e− → A′ and non-resonant e+e− → γA′ annihilation[59, 179];

• Meson decay, π0/η/η′→ γA′, and rare meson decays such as K →πA′,φ→ ηA′, and D∗→ D0A′ [179].

On the other hand, the main methods for DP detection can be broadly sum-marized in bump hunt in visible final-state invariant mass, bump hunt inmissing-mass and vertex detection. In the scenario of an invisible mediatordecay, discussed in the last section, there are two main approaches to ob-serving a hidden sector signature: either a DP is produced at a beam-dumpand promptly decays into invisible states, and the invisible states scatter offmaterial in a detector (placed downstream from the beam-dump), or thepresence of the invisible states is inferred from missing momentum and/ormissing mass either at a high-energy collider or fixed-target experiment.

Eo(e+)

A′

e+

e−

Shield

Detector

Target

Sketch of the setupof a positronbeam-dumpexperiment.

Fixed-target experiments

Fixed-target experiments make use of high-current electron (or positrons)beams13 to search for A′ with masses 2me < mA′ ® GeV and couplings

13 Proton beam fixed-target experiments can also be considered. In this case, an intense pro-ton beam impact on a target producing large numbers of secondary hadrons, most of whichsubsequently decay into SM and weakly coupled hidden-sector particles.


10-2 10-1

mA ′ (GeV)

10-8

10-7

10-6

10-5

10-4

10-3

10-2

εBABAR

(g - 2)e

E137KEK

E774

Orsay

KLOEPADME (e + e − →A ′γ)

E141

FIGURE 5: Dark photons parameter space (visible decay). Limits on the DP ki-netic mixing ε as a function of the mass mA′ from different experiments [59].The regions excluded by the Orsay and KEK experiments are given by the blueand green-yellowish lines, respectively [180]. The (g − 2)e orange line corre-sponds to the constrains from the anomalous magnetic moment of the electron[181, 182], while the green-dashed curve corresponds to the E141 SLAC experi-ment [180, 183]. The hatched region in black is the region that could be excludedby PADME running in the thin target mode [179]. Limit from BaBar searches forA′ → e+e− decays for mA′ > 20MeV are show in red [184]. The lower regionin light gray extending the E137 exclusion limits from the reanalysis in refer-ence [60]. The pink region corresponds to searches for e+e− → γA′ followed byA′→ e+e− in the KLOE experiment [185], while the purple regions are the limitsset by the Fermilab E774 experiment [186].

down to ε ¦ 10−5. In these experiments a large number of charged par-ticles (often electrons) with a fairly high-energy are shot onto a block ora foil (typically of metal). In the interaction between the incident and thetarget particles A′ are produced radiated off by the electrons that scatteroff target nuclei (in a process similar to ordinary bremsstrahlung), or res-onantly produced via e+e− annihilation [59, 179]. The A′, due to theirweak interactions with the SM, can traverse a shield sufficiently long tosuppress the SM background and decay into leptons (mostly e+e− for themass range of interest) which can be detected. Therefore, a decay lengthof O(cm–m) is needed in order for the DP to be observable by decayingbehind the shield and before the detector.The produced electron-positron pairs can be distinguished from those pro-duced in ordinary electromagnetic interactions in two ways:

1. The invariant mass distribution of e+e− pairs produced via an A′ hasa clear peak at the mass mA′;


2. For very low values of ε the produced A′ is very long lived and wetherefore have very displaced vertices.

Depending on the specific experimental set-up with respect to the decaylength of the DP, the possible reach of an experiment is determined notonly by the collected luminosity but also by the choice of the beam energy,the length of the shield, and the distance to the detector. Large values of ε,for which the lifetime is very short, are not accessible since the DP decayswithin the shield. At very small values of ε, the experimental sensitivityis limited by statistics since there are very few DP that will be producedand that decay before the detector [187–189]. Fig. 5 shows the DP pa-rameter space with some limits on the DP kinetic mixing ε as a functionof the mass mA′ from different experiments. Only the visible DP decay isconsidered. For a more detailed description of the current experimentalsituation including the invisible DP decay see ref. [34]. The figure showsthe status of the current limits for DP searches assuming visible A′ decaysinto e+e− pairs with unit branching fraction and suppressed couplings tothe proton. Old data from KEK [190] and ORSAY [191] have been reana-lyzed in reference [180] yielding the limits display in Fig. 5. E141 was ob-tained in [183] from a reanalysis the E141 experiment at SLAC [136]. TheKLOE-2 experiment has searched for e+e− → γA′ followed by A′ → e+e−

[185] setting the KLOE limit shown in the figure. Constrains from theanomalous magnetic moment of the electron are also considered, labeledas (g −2)e [181, 182]. A comparable limit stems from BaBar searches forA′ → e+e− decays for mA′ > 20 MeV [184] which also constrains a largepart of the parameter space.On the other hand, accelerator-based experiments that make use of a lep-ton beam of moderate energy (∼20 GeV) on a thick target or a beam-dumpshow a sizeable sensitivity to a wide area of light DM parameter space. Forexample, in electron beam-dump experiments, an intense primary beamis dumped on a passive thick target followed by a significant amount ofshielding material. In a positron rich environment, such as the electromag-netic shower produced by the interaction of GeV electrons (or positrons)impinging on a thick target (beam-dump), e+ resonant and non-resonantannihilation are two viable DP production mechanisms. In the case of A′ in-visible decays this also constitutes a production mechanisms for two lightDM particles.

In summary, we can considered light pseudoscalar particles (axions andALPs) and new light vector bosons (DP) as well-motivated states that canexplain some important issues as for example the strong C P problem andDM. In the next chapters, we will discuss specific implementations of frame-works involving these type of particles.

3L I N K I N G A X I O N L I K E D A R K M AT T E R T O N E U T R I N OM A S S E S

In this chapter we will present a simple implementation of the “pseudoscalarand neutrino portal”, where ALPs are invoked to solve the DM density andneutrino masses puzzle.

I N T R O D U C T I O N

Besides elegantly solving the strong CP problem, the PQ mechanism maybe also related to the solution of DM and neutrino puzzles by offeringa candidate for CDM, the axion itself, [192–194] and a connection to theneutrino mass generation [195–201]. In the same ways, ALPs, arising fromspontaneous breaking of approximate global symmetries, are also theoret-ically well motivated since these appear in a variety of ultraviolet exten-sions of the SM [38, 43, 44] and, as in QCD axion models, these can makeup all of DM Universe [84], or be a portal connecting the DM particle to theSM sector [202]. Moreover, from the experimental point of view, there areastrophysical phenomena such as the cosmic γ-ray transparency, the x-rayexcess from the Coma cluster and the x-ray line at 3.55 keV that suggestthe for presence of ALPs as possible explanation [151–153, 203–207].In the context of ALP models, the approximate U(1)X continuous symme-try is typically assumed to be remnant of an exact discrete gauge sym-metry such as a ZN symmetry [105–112]. This discrete gauge symmetry The gravity breaks

U(1)X throughPlanck-scalesuppressedoperators, but thatsymmetry isstabilized imposinga ZN .

protects the ALPs mass against large gravity-induced corrections and it canalso be used to stabilize other mass scales present in the theory. For exam-ple, with the aim of generate neutrino masses, in the models implementedin ref. [44, 208], these types of discrete gauge symmetries are used inorder to protect the associated lepton-number-breaking scale, where theALP is used to explain some astrophysical anomalies. In this work, a self-consistent framework of ALP DM1 is builded with the neutrino masses gen-erated via the Inverse Seesaw (ISS) mechanism [45, 46]. For this purposean appropriate ZN discrete gauge symmetries is necessary to protect thesuitable ALP mass, reproducing the correct DM relic abundance as well asto stabilize the mass scales present in the ISS mechanism. In the latter case,

the ISS mass terms are determined -up to some factor- by terms likevnσ

M n−1Pl

,

where n is an integer determined by the invariance of such terms underthe symmetries of the model and some phenomenological constraints, andvσ is the vev of the scalar field σ(x) that spontaneously breaks the globalsymmetry U(1)X and hosts the ALP, a(x).

1 Note that in [44, 208] ALPs are used to explain some astrophysical anomalies and not toaccount for the entire DM abundance.

27

28 L I N K I N G A X I O N L I K E D A R K M AT T E R T O N E U T R I N O M A S S E S

In order to implement the ISS mechanism we extend the SM matter con-tent by introducing nNR

(nSR) generations of SM-singlet fermions NR(SR)

as it is usual. In this work, we consider the minimal number of singletfermionic fields that allows to fit all the experimental neutrino physics:the (nNR

, nSR) = (2, 2), (2,3) and (3, 3) cases [47]. In each case, the ALPs

plays the role of DM candidate. Moreover, for the (2,3) ISS case there isa possibility of having a second DM candidate: the sterile neutrino (theunpaired singlet fermion) [13, 48, 209, 210]. This motivates the construc-tion of models where the DM of the Universe is composed by ALPs andsterile neutrinos, with the latter being generated through the active-sterileneutrino mixing [211] and accounting for a fraction of the the DM relicdensity.As far as phenomenological issues are concerned, since in each frame-work the approximate continuous symmetry should be anomalous withrespect to the electromagnetic gauge group (through an exotic vector-likefermion) instead of being anomalous with respect to QCD (see section 2.1),it is possible to build an effective interaction term involving the axion-likefield, the electromagnetic field strength and its dual [81]. This implies thatthe ALPs may be detected in current- and/or proposed- experiments thatuse the ALP-photon coupling as their main interaction channel to search forALPs [38, 141]. Moreover, considering this particle as the DM candidate,it can be part of the Milky Way DM halo and could resonantly convert intoa monochromatic microwave signal in a microwave cavity permeated bya strong magnetic field [85, 116, 212]. On the other hand, since a largeportion of the parameter space of ALPs (e.g. low masses and couplings) isrelatively unconstrained by experiments (see Fig. 3) because the conven-tional experiments (Helioscopes, Haloscopes and others [141]) are onlysensitive to axion (and axion-like) particles whose Compton wavelengthis comparable to the size of the resonant cavity, it is important to lookfor new search strategies in order to cover other regions of the parameterspace. To reach smaller values of the ALP mass and ALP-photon coupling,a different experimental approach is necessary like the ones associatedwith the ABRACADABRA proposal [156, 213], where a new set of experi-ments was suggested based on either broadband or resonant detection ofan oscillating magnetic flux, designed for the axion detection in the rangema ∈ [10−14, 10−6] eV. It is precisely these kinds of searches that can beused to probe the benchmark regions that we study within the (2,2) and(3,3) ISS cases.

3.1 F R A M E W O R K

The main ingredients of a SM extension in order to link ALPs to neutrinomass generation, and at the same time, to offer an explanation for theDM relic density currently reported by Planck Collaboration [214] are pre-sented in this section.In order to achieve this aim, the SM matter content must to be extended

3.1 F R A M E W O R K 29

with some extra fields. Besides the scalar σ and fermionic SRα and NRβ

fields, an extra electrically charged fermion E (transforming chirally un- The SM fermionsare assumed totransformvectorially underU(1)X so that noelectromagneticanomaly is inducedby the SM fermions.

der U(1)X ) is also added to the SM to make possible the ALPs couplings tophotons, Eq. (2.18), so that the anomaly coefficient is determined by itsU(1)X charge. Therefore, the main role of the new extra charged fermionsit is to induce an ALPs coupling to two photons. As a consequence, theALPs can be found, in principle, in current and/or proposed experimentsthat make use of the ALP-photon coupling (see discussion in section 2.1.5).Another key point of the framework is the existence of a ZN discrete gaugesymmetry. In order to understand its role let’s consider:

1. Imposing an anomalous U(1)X symmetry to the Lagrangian doesnot seem sensible in the sense that in the absence of further con-straints on very high energy physics, it is expected that all admissi-ble and marginally relevant operators that are forbidden only by thissymmetry will appear in the effective Lagrangian with coefficientof order one. However, if this symmetry follows from some otheranomaly-free symmetry, in this case a ZN discrete gauge symmetry,all terms which violate it are then irrelevant in the renormalizationgroup sense.

2. The ZN symmetry protects both the ALPs mass and the ISS textureagainst gravity effects.

For these reasons, the effective Lagrangian in the models will be invariantunder a ZN discrete gauge symmetry. Due to the fact that ALP mass is verysmall and only protected by the U(1)X symmetry (which is explicitly bro-ken by gravity effects through the non-renormalizable operators), the ZN

symmetry will have a high order N 2. This also happens in models withQCD axions and it is shared by all models with this type of stabilizationmechanism [44, 100, 208, 215–217].

3.1.1 Lagrangian

The effective Lagrangian that we consider to relate the ISS mechanism toALPs DM is composed of four terms:

L ⊃ LYukSM +Lσ+LISS +LE , (3.1)

where LYukSM represent the Yukawa Lagrangian of the SM, eH = iτ2H∗ with τ2

the second Paulimatrix.LYuk

SM = Y (u)i j QLi eHuR j + Y (d)

i j QLiHdR j + Y (l)i j LiHlR j + h.c., (3.2)

with the usual QLi , uRi , dRi and Li , lRi fields denoting the quarks and lep-tons of the SM, respectively, and with H being the Higgs SU(2)L doublet.The others three terms in Eq. (3.1) are discussed as follow.


3.1.1.1 The Lσ term

The Lσ term, the Lagrangian involving the σ field, which is relevant in ourdiscussion represents the nonrenormalizable operators given in Eq. (2.21),that is:

Lσ ⊃ gσD

M D−4Pl

+ h.c., (3.3)

where g = eiδ |g| and D is an integer. The σ field is parametrized as:

σ(x) =1p

2[vσ+ρ(x)]ei a(x)

vσ , (3.4)

where a(x) is the ALP field and ρ(x) the radial part that will gain a massof order the vev, vσ ≡

p2 ⟨σ⟩, which can take values in the following range

[85, 87–89, 93]:

109 ® vσ ® 1014 GeV. (3.5)

In section 2.1.3 it was shown that with these non-renormalizable operatorsit is possible to obtain the ALPs mass term given in Eq. (2.23), which canbe written as:

ma = |g|12 DMPlλ

D2 −1, (3.6)

where λ≡ vσp2MPl

can take values in the range 10−10 ® λ® 10−5 accordingto Eq. (3.5).

3.1.1.2 The LISS term

Regarding the neutrino mass generation, once we introduce the NRβ andSRα fields to implement the ISS mechanism, the LISS Lagrangian is givenby:

LISS = yiβ Li eHNRβ + ζαβσp

M p−1Pl

SRα(NRβ )C +ηαα′

σq

2Mq−1Pl

SRα(SRα′)C

+θββ ′σr

2M r−1Pl

NRβ (NRβ ′)C + h.c., (3.7)

where the coupling constants yiβ , ζαβ ,ηαα′ , θββ ′ , with i, j = 1,2, 3,α,α′ =1, 2, (or 3) and β ,β ′ = 1,2, (or 3), are generically assumed of order one.The exponents p, q, r are integer numbers chosen for satisfying some phe-nomenological constraints discussed below. Without loss of generality, theNegative values for

the p, q, r exponentswill mean that the

term is ∼ σ∗n

instead of ∼ σn.

exponent p can be assumed to be positive. At this point we only con-sider the minimal number of neutral fermionic fields, SRα and NRβ , thatallow to us fit all the experimental neutrino physics. In particular, the(nNR

, nSR) = (2, 2), (2,3) and (3,3) cases are studied [47].

3.1 F R A M E W O R K 31

As the σ field gets a vev, the gravity-induced terms in Eq. (3.7) give themass matrix for light (active) and heavy neutrinos [208]. In the basis(νL, NC

R , SCR) the mass matrix can be expressed as:

Mν =

0 MᵀDMD MR

, (3.8)

with MD ≡

mD

0

and MR ≡

µN Mᵀ

M µS

, (3.9)

where mD, M , µN and µS are matrices with dimension equal to nNR× 3,

nNR×nSR

, nNR×nNR

, nSR×nSR

, respectively. The energy scales of the matrixelements are determined essentially by

p2 ⟨H⟩ ≡ vSM ' 246 GeV, λ (or vσ)

and MPl as follows:

mD iβ = yiβvSMp

2, Mαβ = ζαβMPlλ

p, (3.10)

µSαα′ = ηαα′MPlλ|q|, µN ββ ′ = θββ ′MPlλ

|r|. (3.11)

The mass matrix in Eq. (3.8) allows light active neutrino masses of theorder of sub-eV without resorting to very large energy scales, in contrast Making µS and µN

small is technicallynatural, in the senseof µN ® µS mD < M .

to the type I seesaw mechanism [218–223]. Assuming the hierarchy µN ®µS mD < M and taking a matrix expansion in powers of M−1, the lightactive neutrino masses, at leading order, are approximately given by theeigenvalues of the matrix [224, 225]

mνlight ' mᵀDM−1µS(Mᵀ)−1mD. (3.12)

A crucial point to remark here is that this expression is only valid in scenar-ios of squared matrices, such as the (2,2) ISS and (3,3) ISS cases describedbelow. In other cases, as e.g. the (2,3) ISS, the light active neutrino masseshas the form [47, 48]:

mνlight 'O(µ)O(k2), (3.13)

where k =O(mD)O(M) . On the other hand, the heavy neutrino masses are given In this case, the

mixing between theheavy neutrinosand the light activeneutrinos ismodulated by theratio mD M−1

[225].

by the eigenvalues of the matrix MR,

mνheavy 'O(M)+O(md) ∼O(M). (3.14)

Note, from Eq. (3.12), that µN does not contribute to the light active neu-trino masses at the leading order [224, 225], although ti contributes at oneloop. Actually, the presence of µN term gives a sub-leading contribution tomνlight matrix at order of mᵀDM−1µS(Mᵀ)−1µN M−1µS(Mᵀ)−1mD, which isa factor µSµN

M2 smaller than the leading contribution [208].Well-motivated scales for M and µS , µN are TeV and keV scales, respec-tively. These scales allow getting active neutrino masses in the sub-eVrange without considering smaller Yukawas and, in some scenarios, such


as the (2, 3) ISS case, allow the existence of a keV sterile neutrino as awarm dark matter (WDM) candidate [48]. In addition, M has to satisfy:

M ¦s

10µS

keVTeV, (3.15)

because light active neutrino masses are at the sub-eV scale and mD is oforder O(vSM). Another important constraint on the M scale comes fromthe fact that the mixing matrix that relates the three left-handed neutrinoswith the three lightest mass-eigenstate neutrinos is no longer unitary. Thisimplies that deviations of some SM observables may be expected, suchas additional contributions to the `νW vertex and to lepton-flavor- andCP-violating processes, and nonstandard effects in neutrino propagation[226, 227]. For example, in the ISS model, the violation of unitary is oforder of ε2, with ε ≡ mDM−1 being approximately the mixing betweenthe light active and heavy neutrinos [225]. Roughly speaking, ε2 at thepercent level is not excluded experimentally [226, 228–230].Taking into account the previous considerations, the best ranges chosenfor M and µS scales are:

1 TeV≤ M ≤ 25 TeV and 0.1 keV≤ µS ≤ 50 keV. (3.16)

Once the scales of the mass matrices are established and using Eqs. (3.10)and (3.11) (following a similar procedure as in ref. [208], where the effec-tive values of p and q parameters that satisfy the constraints over vσ andthe appropriate scales for the ISS mechanism are studied), the integers pand q in Eq. (3.7) can only take the values

(p, |q|) = (2,3) for 6×1010 ®vσ

GeV® 1×1011, (3.17)

(p, |q|) = (3, 5) for 2×1013 ®vσ

GeV® 8×1013, (3.18)

where the vσ range corresponds to the interval where all the above con-straints are satisfied. That happens because the same vev simultaneouslyprovides the M and µS scales. Besides the constraints over p and q, the ex-ponent r in the last expression of Eq. (3.7), which generates the µN term,is also constrained to be r ≥ |q| because µN must be lower than µS .Notice that for both possibilities in Eqs. (3.17) and (3.18) the light activeneutrino mass matrix in Eq. (3.12) is simplified, using Eqs. (3.10) andThe active neutrino

masses areindependent of thePlanck scale and itsvalue is determinedby the vσ parameter.

(3.11), to:

mνlight =

yᵀζ−1η (ζᵀ)−1 y v2

SMp2vσ

, (3.19)

where we have suppressed the α, β index for simplicity.

3.1.1.3 The LE term

Finally, in order to consider every element present in Eq. (3.1), LE in-volves the terms associated with the new charged lepton, E, necessary to

3.1 F R A M E W O R K 33

generate a non-null gaγγ coupling. As was discussed in section 2.1.3, thisALP-photons coupling is determined by the interaction term in Eq. (2.17),anomaly induced and, at leading order, given by Eq. (2.18). 2 Notice here

a

E

γ

γ

E

E

gaγγ

ALP-photoncoupling.

that the existence of a non-null anomaly coefficient, Eq. (2.19), in thisexpression guarantees that gaγγ 6= 0. This is the reason why the total La-grangian in Eq. (3.1) is invariant under an anomalous U(1)X global sym-metry. Nevertheless, only with SM fermions and the neutral SRα and NRβ

fermions is not possible to have an anomalous U(1)X symmetry in the elec-tromagnetic group. Therefore, we need to include a new SU(2)L singletfermion, E, with one unit of electric charge. This new field can be writtenas:

LE ⊃ ϑiσs

M sPl

LiHER +κσt

M t−1Pl

ELER +H.c., (3.20)

where ϑi and κ are Yukawas, in principle, assumed of order one. These twoterms are also subjected to phenomenological and theoretical constraintsas follow.

• The term ∼ σt ELER must give a mass large enough for the E leptonto satisfy the experimental constraints over any new exotic chargedleptons. That is, mE > 102.6 GeV at 95% C.L. for stable chargedheavy lepton [74], and mE > 574 GeV at 95% C.L. for chargedlong-lived heavy lepton, assuming a mean life above (7× 10−10 −3× 10−8) s [231, 232]. These constraints are satisfied if the t pa-rameter is less or equal than 3. Moreover it must be different fromzero because the electromagnetic anomaly must be present.

• The s parameter can take the values 1 or 2 because∼ σs LHER deter-mines the interaction of E with the SM leptons, and if s is larger than2 the charged E lepton becomes stable enough to cause cosmologicalproblems.

3.1.2 ZN symmetry anomaly cancellation

The gravitational effects must be controlled to give a suitable ALPs mass.With this aim, a gauge discrete ZN symmetry is introduced assumed tobe the remnant of a gauge symmetry valid at very high energies [105].Thus, to protect the ALPs mass against the gravitational effects, the ZN

symmetry must at least be free from anomalies [106–109], i.e., the ZN

charges of the fields must satisfy the following equations (see appendix Afor more details):

A2(ZN ) = A3(ZN ) = Agrav(ZN ) = 0 ModN2

, (3.21)

2 Higher corrections to the gaγγ coupling are possible (for an extensive study of them see ref.[128]). However, for the suitable ALPs masses in the range that can explain the observedDM relic density, all of them can be safely neglected.


QLi dRi uRi Hi Li lRi NRβ SRα EL ER σ

B 13

13

13 0 0 0 0 0 0 0 0

L 0 0 0 0 1 1 1 a b c d

TABLE 1: Assignment of charges for two of the continuous symmetries presentin the Lagrangian of Eq. (3.1). The charges a, b, c and d are given bya = qd/2, b = sd + c, c = 1− rd and d = (p− q/2)−1 from L (leptonnumber) conservation in the model.

where A2, A3 and Agrav are the [SU(2)L]2× ZN , [SU(3)C]

2× ZN and[gravitational]2× ZN anomalies, respectively. Other anomalies, such as Z3

N ,do not give useful low-energy constraints because these depend on somearbitrary choices concerning the full theory.Gravitational effects can also generate terms such as:

σn

M n−1Pl

SRSCR ,

σn

M n−1Pl

SRNCR ,

σn

M n−1Pl

NCR NR,

σn

M nPl

L eHSR, (3.22)

with n smaller than those in the Lagrangian of Eq. (3.7) that jeopardizeboth the mass matrix structure in Eqs. (3.8) and (3.9), and the scales ofthe ISS mechanism. Thus, the ZN will be chosen such that it also preventsthese undesirable terms from appearing.In general, the ZN symmetry can be written as a linear combination ofthe continuous symmetries present in the model: the hypercharge Y , thebaryon number B, and the generalized lepton number L. The charge as-signments for B and L symmetries are shown in table 1, whereas the as-signment for the Y symmetry is the canonical one in the SM. The chargesfor the L symmetry are chosen considering the conservation of the ex-tended lepton number (similar to case in ref. [208]). Considering the sym-metries present in table 1, since the hypercharge is anomaly free by con-struction, the ZN charges (Z) of the fields can be written as Z = c1B+ c2L,where c1,2 are rational numbers in order to make the ZN charges integers.This assignment is useful in order to solve the anomaly equations (3.21) tosatisfy the anomaly free condition. Substituting the charges in table 1 intothe equation Z = c1B+ c2L (see refs. [106–109]) it is possible to obtainthe following anomaly coefficients (see appendix A):

A3(ZN ) = 0, (3.23)

A2(ZN ) =32[c1 + c2] , (3.24)

Agrav(ZN ) = c2

h

3− nNR− nSR

qd2

+ sdi

. (3.25)

From these equations, A2(ZN ) and Agrav(ZN ) are not, in general, 0 ModN/2. This implies strong constraints on the choice of the ZN discrete sym-metry, because all these equations must be equal to 0 Mod N/2 in orderto consider ZN an anomaly free discrete gauge symmetry.

3.1 F R A M E W O R K 35

3.1.3 ALPs and sterile neutrino dark matter

Since the ALPs are very weakly interacting slim particles and cosmologi-cally stable, they can be a DM candidate [84] (see discussion in section2.1.4). In fact, ALPs may be non-thermally produced via the misalignmentmechanism in the early Universe and survive as a cold dark matter popu-lation until today, with the relic density determined from the expressionin Eq. (2.25), where the initial misalignment angle, Θi , is taken as πp

3,

because we are assuming a post-inflationary symmetry-breaking scenario,which is favorable for models with vσ ® 1014 GeV [84, 85].On the other hand, as was commented in the introduction of this section,there is a configuration in the ISS mechanism −(2,3) ISS− where a sterileneutrino can be considered as a DM candidate. This is treated as a WDMcandidate whose relic density can be generated through the Dodelson-Widrow (DW) mechanism [211], which is present as long as active-sterilemixing is not zero [13, 209, 210]. In this case, the fraction of DM abun-dance depends on the sterile neutrino mass, mνS

, and its mixing angle withthe light active neutrino θ .In the (2, 3) ISS case, the sterile neutrino through the DW mechanism canaccount at most for≈ 43% of the observed relic density without conflictingwith observational constraints [48].3 Since its relic density depends on themass, the following two considerations must be taken in to account:

1. For mνS> 0.1 keV, the relic density produced by the usual DW mech-

anism is given by [48, 233]:

ΩνS ,DMh2 = 1.1×107∑

α

Cα(mS)|UαS|2mνS

keV

2

, (3.26)

with α = e, µ, τ, and where Cα(mS) are flavor-dependent coeffi-cients which are calculated by solving numerically the Boltzmannequations. An appropriate value, for the models considered in thiswork, is Cα(mS) ' 0.8 (for more details see ref. [233]). In Eq. (3.26),the sum of UαS (the elements of the leptonic mixing matrix) is theactive-sterile mixing, i.e.,

∑

α |UαS|2 ∼ sin2(2θ ).

2. For mνS< 0.1 keV, with the above definition of the active-sterile

mixing, the relic density can be written as [48, 234]:

ΩνS ,DMh2 = 0.3

sin2 2θ10−10

mνS

100 keV

2

. (3.27)

After imposing bounds coming from stability, structure formation and in-direct detection (in addition to the constraints arising from the neutrinos

3 This DM amount can be slightly increased to≈ 48% by including some effect of the entropyinjection of the pseudo-Dirac neutrinos, provided the lightest pseudo-Dirac neutrino hasmass between (1−10) GeV (see ref. [48] for more information).


oscillation experiments), we find that the sterile neutrino as WDM in the(2, 3) ISS case provides a sizable contribution to the DM relic density for:

2 keV® mνS® 50 keV and 10−8 ® sin2(2θ ) ® 10−11, (3.28)

where the maximal fraction of DM is achieved when the sterile neutrinohas a mass mνS

' 7 keV [235–238] (see ref. [48] for more detail).Once we have established the DM candidates and the parameters that de-termine the relic density in each case, we are going to search for modelssatisfying all of the conditions mentioned in section 3.1 as well asA multicomponent

DM model withALPs and sterile

neutrinos as the DMcandidates.

ΩPlanckDM h2 = ΩνS ,DMh2 +Ωa,DMh2, (3.29)

where ΩPlanckDM h2 = 0.1197± 0.0066 (at 3σ) is the current relic density as

reported by the Planck Collaboration [214].

3.2 M O D E L S

In the previous section, we have introduced the general and minimal con-straints that the models must satisfy. In this section, we proceed to findspecific models that give an explanation to the DM observed in the Uni-verse. In particular, the (2,2), (3,3) and (2,3) cases of the ISS mecha-nism are studied in detail. In each model we check the compatibility (at3σ) with the experimental neutrino physics for the normal mass order-ing [227] and vanishing C P phases by varying the free Yukawa couplingsyiβ ,ζαβ ,ηαα,θββ in the range ∼ (0.1,3.5). Additionally, we also analyzethe lepton-flavor-violating (LFV) processes such as `β → `α+γ, which areinduced at one loop by the W boson and the heavy neutrinos, with:

Br(`β → `αγ) =α3

W s2W

256π2

m5`β

m4WΓ`β

∑

i

U∗β iUαiG(m2Ni

/m2W )

2

, (3.30)

W

γ

lαlβ ν

LFV processes. where G(x) = x(1− 6x + 3x2 + 2x3 − 6x2 log(x))/[4(1− x)4], Γ`β isthe total decay width of `β and U represents the lepton mixing matrix[239, 240]. Each ISS model is checked to be compatible with the currentexperimental limits Br(µ→ eγ) < 5.7× 10−13, Br(τ→ eγ) < 3.3× 10−8,and Br(τ→ µγ) < 4.4×10−8 [241–243].

3.2.1 (2,2) ISS case

Among the minimal configuration of the ISS mechanism consistent withthe experimental neutrino physics and LFV processes [47, 48, 244], wefirst study the (2,2) ISS case because this is the minimal configuration thatsatisfies all the constraints coming from experimental neutrino physics. Forthis case, in the neutrino mass spectrum, there are two heavy pseudo-Diracneutrinos with masses M ∼ O(TeV), two light active neutrinos with sub-eV masses coming from the mass matrix in Eq. (3.12), and one massless

3.2 M O D E L S 37

D vσ (GeV) vσ (GeV)

8 (1.9−3.3)×1010 (1.5−2.8)×1010

9 (0.6−1.1)×1011 (5.5−9.5)×1010

10 (1.9−3.2)×1011 (1.6−2.8)×1011

11 (5.0−8.2)×1011 (4.4−7.2)×1011

12 (1.2−1.9)×1012 (1.0−1.7)×1012

13 (2.6−4.0)×1012 (2.3−3.6)×1012

14 (5.2−7.9)×1012 (4.6−7.0)×1012

15 (1.0−1.4)×1013 (0.9−1.3)×1013

16 (1.7−2.5)×1013 (1.6−2.3)×1013

17 (2.9−4.2)×1013 (2.7−3.8)×1013

18 (4.8−6.7)×1013 (4.3−6.1)×1013

19 (0.7−1.0)×1014 (6.8−9.4)×1013

TABLE 2: (D, vσ) values that can provide the total (second column) and partial(57 % in the third column) DM relic density by the ALP.

eigenstate to all orders in perturbation theory [47]. Because in this casethe number of extra singlet fermions is nNR

= nSR= 2 (similarly for the

(3, 3) ISS case), there is no sterile neutrino νS in the mass spectrum, whichis only present when nNR

6= nSR. Therefore, all the DM abundance must be

constituted by ALPs, i.e.,

ΩPlanckDM h2 = Ωa,DMh2. (3.31)

In order to find the main features of the model, it is useful to rewriteΩa,DMh2 in terms of the ALP mass ma and D, that is, the exponent of themass operator for σ in Eq. (3.3). Thus, substituting Eq. (3.6) in Eq. (2.25),considering the misalignment angle θ2

a∼= π2/3, we find that:

Ωa,DMh2 ' 0.49 |g|14p

D exp

−D4

ln

p2MPl

1 GeV

h vσ1 GeV

iD+6

4, (3.32)

where g is assumed to be 10−3 ≤ |g| ≤ 2. From this equation, we can seethat Ωa,DMh2 only depends on the parameters (|g| , D, vσ). In table 2 weshow the allowed (D, vσ) values for the cases where ΩaDMh2 = ΩPlanck

DM h2

and Ωa,DMh2 = 0.57×ΩPlanckDM h2, that is, when the relic density DM is ex-

plained totally or partially by the ALP, respectively. In this table, the lastcase applies only for the (2,3) ISS case and will be discussed in section3.2.3, therefore in this section only the first two columns are considered.


With the parameters in table 2, in order to obtain the Lagrangian for thisscenario, we search for discrete symmetries according to the two possibili-ties shown in Eqs. (3.17-3.18) and different values of r, s, t with the follow-ing constraints: considering Eqs. (3.17-3.18) and table 2 we can see that,for the range of values to vσ established in section 3.1.1.1, only the valuesD = 9,10, 16−19 satisfies the general and minimal constraints describedin the previous sections to obtain a model that can explain the neutrinophysics and the correct relic density with ALPs. Thus we search for discretesymmetries ZN that give rise to mass operators with dimensions D, withthe following results:

1. The Z9,10, symmetries allow terms such as:

σ∗

MPlL eHSR,

σ∗2

MPlNRN C

R , L eHSR, (3.33)

and since H and σ get vevs, these terms do not give the appropriatezero texture of the ISS mechanism shown in Eqs. (3.8) and (3.9). Forthis case, we have also searched for all of the possible combinationsof r, s, t values in the Lagrangian of Eq. (3.7) without any success.

2. The Z16,18 symmetries are not free of the gravitational anomaly. Infact, the ZN≤20 discrete symmetries that satisfy all of the anomalyconstraints and stabilize the ISS mechanism are Z17,19.

Z17 symmetry

For the Z17 symmetry one solution is found. For this case, the Lagrangianof the model, LZ17

, is given by Eq. (3.1) with the parameters

D = 17, (p, q, r) = (3,−5,−6), and (s, t) = (2,2) (3.34)

in Eqs. (3.3), (3.7) and (3.20), respectively. An assignment of the Z17

charges and the anomalous U(1)X symmetry for this model is shown intable 3. Notice here that, from the discussion in section 3.1.1.2, the term∼ σ∗6NRβ (NRβ ′)

C in Eq. (3.7) gives a negligible contribution for the lightactive neutrino masses, which does not jeopardize the mass matrix texturein the ISS mechanism.Regarding both ALPs and neutrino phenomenology, respectively, we havefound that:

1. The benchmark region in this model is denoted as M22a in Fig. 6,where we can see some excluded regions from various experimentsand the other benchmark regions for the different models describesbelow. These values for gaγγ and ma allow that the ALPs explain100% of the DM density considering the values 10−3 ≤ |g| ≤ 2and 2.9× 1013 ® vσ

GeV ® 4.2× 1013. In the particular case of vσ ∼=

3.2 M O D E L S 39

Axion

CDM

Massive Stars

SN1987A γ-Ray Burst

γ-Ray TransparencyIAXO

ALPS-II

Halo

scopes

3.55keV

Linefrom

DecayingALPDM

ALPCDM

M22a.

M22b.

M33a.M33b.

M23a.M23b.

ABRA-Res.

ABRA-Broad.

-12 -10 -8 -6 -4 -2 0 2 4

-18

-16

-14

-12

-10

-8

Log10 ma [eV]

Log

10|g

aγ|[

GeV-

1]

FIGURE 6: ALPs parameter space (benchmark regions). Some excluded regionsfrom anomalous energy loss of massive stars, γ-ray burst from SN 1987A, andof DM axions or ALPs converted into photons in microwave cavities are shown.Green regions are the projected sensitivities of experiment like ALPS-II, IAXO,ADMX and ADMX-HF. The black (green) solid line in the lower left corner showsthe sensitivity of the proposed ABRACADABRA experiment. The QCD axion re-gion is shown in the yellow band. The benchmark regions M22a, M22b, M23a,M23b, M33a and M33b correspond respectively to models (2,2) ISS, (2,3) ISSand (3, 3) ISS which generate a considerable amount of the DM relic density.

3.08× 1013 GeV, from Eqs. (3.6) and (2.18), gaγγ and ma are givenby:

gaγγ∼= 7.54×10−17

3.08×1013 GeVvσ

GeV−1, (3.35)

ma∼= 5.59×10−10|g|

12

h vσ3.08×1013 GeV

i15/2eV.

2. Sharp predictions for neutrino masses are not possible with just theknowledge of the p, q, r, s, t values and vσ. However, the order ofmagnitude of the mass matrices can be estimated from Eqs. (3.10)and (3.11) to be:

M ∼= ζ×1.73 TeV, µS∼= η×0.13 keV,

mνlight '

yᵀζ−1η (ζᵀ)−1 y

×1.38 eV, (3.36)

where the value vσ ∼= 3.08× 1013 GeV has been used. These massscales are appropriate to satisfy the constraints coming from experi-mental neutrino physics and unitarity without resorting to fine tun-ing in the couplings. Nevertheless, some care must be taken in order


Model Symmetry QLi dRi uRi H Li lRi NRβ SRα EL ER σ

(2,2)

Z17 1 2 0 16 14 15 13 8 15 12 7

U(1)X 0 0 0 0 11/2 11/2 11/2 −5/2 11/2 15/2 1

Z19 1 14 7 6 16 10 3 9 17 2 4

U(1)X 0 0 0 0 11/2 11/2 11/2 −5/2 5/2 7/2 1

(3,3)

Z17 1 10 9 8 14 6 5 7 2 15 4

U(1)X 0 0 0 0 11/2 11/2 11/2 −5/2 9/2 7/2 1

Z19 1 13 8 7 16 9 4 12 9 11 18

U(1)X 0 0 0 0 11/2 11/2 11/2 −5/2 11/2 7/2 1

(2,3)

Z10 1 2 0 9 7 8 6 6 8 6 6

Z4 0 1 3 3 0 1 3 1 1 1 2

U(1)X 0 0 0 0 7/2 7/2 7/2 −3/2 7/2 3/2 1

TABLE 3: Discrete and continuous charge assignments of the fields in the differentmodels.

to generate the benchmark region M22a in agreement with boundscoming from LFV processes such as µ → e + γ. Specifically, due tomNi∼ M mW , the loop function tends to G(x) → 1/2 and the

mixing terms are generically given by U ∼ mD/M . This leads to thedecay rate for µ→ e+ γ of order

Br(µ→ eγ) ∼ 1.1×10−13 mD

10GeV

43 TeVM

4

, (3.37)

which implies that small y ∼ 0.1 couplings must be required.

Z19 symmetry

For the case with Z19, another solution is found. An effective Lagrangiancan be built with the parameters (p, q, r, s, t) = (3,−5,−8,−2,1) in Eqs.(3.7) and (3.20). The results, roughly speaking, are quite similar to theprevious model with Z17, in the sense that, because the p, q, and |s| valuesare equal for both models (check Eq. (3.34)), the neutrino spectrum issimilar in both cases. Nevertheless, since D, vσ and t have not equal values,the ALPs mass, the mass term for the exotic fermion E and the ALP-photoncoupling, gaγγ, are different (see table 4).

3.2 M O D E L S 41

The benchmark region in this model corresponding to the case in Fig. 6is denoted as M22b. These values for gaγγ and ma also allow to explainthe 100% of the DM density considering vσ ∈ (0.7− 1.0)× 1014 GeV. In The upper bound

on µ→ e+ γ, iseasily fulfilled dueto the largersuppression comingfrom M ∼ 50 TeV(check table 4).

the particular case where vσ ∼= 1.04× 1014 GeV the ALPs parameters aregiven by:

gaγγ∼= 1.12×10−17

1.04×1014 GeVvσ

GeV−1, (3.38)

ma∼= 1.87×10−10|g|

12

h vσ1.04×1014 GeV

i17/2eV.

3.2.2 (3,3) ISS case

In the (3, 3) ISS case there is no light sterile neutrino in the mass spectrum(because nNR

= nSR= 3), all the three active neutrinos are massive and

there are three heavy pseudo-Dirac neutrinos whit masses of order of TeV.Therefore, all of the DM abundance in this model has to be made of ALPs.Proceeding in a similar manner to the (2,2) ISS case, taking into accountthat Agrav(ZN ) is now different because the number of extra new singletfermions is larger (see Eq. (3.25)), we search for all anomaly-free ZN sym-metries with N ≤ 20, and with (p, q, r, s, t) values established accord-ing to the constraints imposed in section 3.1.1. The following results arefound.4

1. The Z9 symmetry is not free of gravitational anomalies, while the Z10

symmetry allows dangerous terms such as L eHSR, σ∗

MPlL eHSR, σNRN C

R ,and others that jeopardize the matrix structure in Eqs. (3.8) and(3.9). Therefore, it is not possible to build a model for the solutionin Eq. (3.17).

2. The Z16,18 symmetries corresponding to the solution in Eq. (3.18)are not free of gravitational anomalies. Therefore these are not suit-able symmetries.

3. The Z17 symmetry forbids the dangerous terms and allows to buildan effective Lagrangian characterized by the parameters D = 17 and(p, q, r, s, t) = (3, −5, 7, 2, 1) in Eqs. (3.3), (3.7) and (3.20), re-spectively. Notice here that this Lagrangian is very similar to the(2, 2) ISS Lagrangian. However, in this case, the exponent of themass term for the exotic lepton E is equal to one and the term asso-ciated with µN is not allowed with dimension less than seven. Sincethe mass term for the exotic leptons is different from (2,2) ISS model,the anomaly coefficient Caγγ is different. We can see this from thecharges in table 3 and Eq. (2.19). Therefore, the ALP-photon cou-pling has also a different value.

4 An effective Lagrangian for the (3,3) ISS case was worked in ref. [208] with the aim ofexplaining some astrophysical phenomena. However, in that case, the DM abundance viaALPs was not considered.


Regarding the ALPs phenomenology, a benchmark region for thiscase is denoted as M33a in Figure 6, where the ALP can explainagain the 100% of the DM relic density. In particular, since vσ hasthe same value than in the (2, 2) ISS case, ma is equal to the valuegiven in Eq. (3.35). Instead, gaγγ turns out to be:

gaγγ∼= 3.77×10−17

3.08×1013 GeVvσ

GeV−1, (3.39)

because the anomaly coefficient now has a different value.

On the other hand, since the parameters (p, |q|) and vσ are equal inboth cases, the neutrino spectrum is the same as in the M22a model(see Eqs. (3.10-3.11)).

4. For the Z19 case, it is found that the model is determined by theparameters (p, q, r, s, t) = (3, −5, −8, 2, 2), D = 19, and the vσvalues given in table 2. This configuration leads to similar conclu-sions as in the M22b model, with some differences coming from theanomaly coefficient Caγγ since the mass term for the exotic leptonsdiffers from the M22b model (check the charges in table 3). Thisimplies that the ALP-photon coupling is:

gaγγ∼= 2.24×10−17

1.04×1014 GeVvσ

GeV−1. (3.40)

The other parameters associated to the neutrino spectrum and ma

are similar to the M22b case, and are shown in table 4. In this case,the benchmark region to explain the 100% of relic DM is denoted asM33b in Figure 6.

On the other hand, the constraints and prospects regarding LFV pro-cesses are similiar to those in the (2,2) ISS case since the mass scaleM of the benchmark regions M33a and M33b is the same as that inthe benchmark regions M22a and M22b, respectively (see table 4).

3.2.3 (2,3) ISS case

For this case, since there are nNR= 2 and nSR

= 3 neutral fermions,the neutrino mass spectrum contains two heavy pseudo-Dirac neutrinoswith masses M ∼ O(TeV), two light active neutrinos with sub-eV massesand, similar to the (2,2) ISS model, there is also a massless state in thisconfiguration [47]. In addition, there is a sterile neutrino, νS , with massµS ∼ O(keV) which can be considered as a DM candidate [48]. Then, forthis model, the presence of both the νS and the ALPs, a, brings the possi-bility of having two DM candidates.First, let’s consider the case ΩPlanck

DM h2 = Ωa,DMh2, i.e., when the DM abun-dance is totally constituted by ALPs.

3.2 M O D E L S 43

3.2.3.1 ALPs as a DM candidate

From Eqs. (3.17) and (3.18) and table 2, is possible to see that:

(D, vσ) =

9, (0.6−1.1)×1011 GeV

, (3.41)

(D, vσ) =

10, (1.9−3.2)×1011 GeV

,

corresponds to the (p, |q|) = (2,3) solution in Eq. (3.17) (note that thevσ value corresponding to D = 10 is slightly out of the allowed range inEq. (3.17)). Moreover, as was discussed in previous sections, the valuesD = 9,10 restrict the discrete symmetry to be Z9,10, such that is necessaryto search for anomaly free solutions for these symmetries, i.e., solutions toEqs. (3.24) and (3.25) with (p, |q|) = (2,3). Nevertheless, all the solutionsfor the Z9,10 charges allow terms as:

σNRβ (NRβ ′)C,σ∗2

MPlNRβ (NRβ ′)

C,σ∗

MPlLi eHSRα, (3.42)

and others that do not give the correct texture to the mass matrix in theISS mechanism, even with all the possible combinations of the r, s and tvalues in the Lagrangian of Eq. (3.1). Therefore, the (p, |q|) = (2,3) casecannot offer a realization of an effective model providing all the observedDM abundance via ALPs when all the constraints in section 3.1 are consid-ered. However, from table 2, the possibilities with D = 15, . . . , 19 (with alarger value of vσ) provide a second solution to the case (p, |q|) = (3, 5)according to Eq. (3.18). 5

Studying the symmetries Z15−19 we have found that Z17 and Z19 are ex-cluded because the condition for the gravitational anomaly is never satis-fied, while in the Z16,18 cases, terms as∼ Li eHSRα and∼ σNRβ (NRβ ′)

C givean incorrect texture for the ISS mass matrix. In fact, after imposing all theconstraints, the only symmetry that provides a solution is Z15.

Z15 symmetry

This model is characterized by the effective Lagrangian, LZ15, given by Eqs.

(3.3), (3.7) and (3.20) with the parameters D = 15 and (p, q, r, s, t) =

(3,−5,−4, 2, 2), respectively. The Lagrangian is also invariant under aU(1)X symmetry anomalous in the electromagnetic group as must be togenerate a non-null coupling gaγγ between photons and ALPs. In this model,the ALPs parameters are given by:

gaγγ ' 2.25×10−16

1.03×1013 GeVvσ

GeV−1, (3.43)

ma ' 4.47×10−8|g|12

h vσ1.03×1013 GeV

i13/2eV.

5 Notice here that, strictly speaking, the vσ value corresponding to D = 15 is slightly out ofthe allowed range in Eq. (3.18).


The neutrino mass spectrum is:

M ' ζ×6.5×10−2 TeV; µS ' η×5.8×10−4 keV;

mνlight ' 4.15×O(y2)O(η)

O(ζ2)eV, (3.44)

where we have used the particular value vσ ' 1.03× 1013 GeV, which isone of the suitable values given in table 2 for D = 15, giving the 100% ofthe current DM abundance. For this case, the sterile neutrino as DM can-didate has a negligible contribution because the small scales in Eq. (3.44)imply that the mixing angle between the active and sterile neutrinos has agreat suppression. Moreover, the mass scale of the sterile neutrino, µS , isvery small to bring a considerable contribution to DM.At this point it is important to say that, from the values of M , µS , mνlight

in Eq. (3.44) it is possible to see that in this scenario there is some tensionfor satisfying the unitarity constraint. In more detail, let’s consider

yζ

<M

vSM×10−1 = 2.6×10−2, (3.45)

where the conservative choice ε2 = 1% to the unitarity has been made

(recall that ε ≡ mDM−1). However, this upper bound on

yζ

implies a

lower bound on η, that is,

η >

yζ

−2 mνlight

4.15≈

yζ

−2Æ

∆m2atm

4.15≈ 17.17, (3.46)

with ∆m2atm = 2.32× 10−3 eV2 being the atmospheric squared-mass dif-

ference. From Eq. (3.46) it is deduced that there is no a perturbative valuefor η. This happens because the values for vσ corresponding to D = 15 issmaller than the values allowed in the range of Eq. (3.18). Similar conclu-sions are found when the case Ωa,DMh2 < ΩPlanck

DM h2 is considered, that is,when the ALPs can only explain a fraction of the relic density. Therefore,the effective Lagrangian LZ15

can not provide a natural framework for DMand the neutrino masses in the (2,3) ISS case. For this reason all the pos-sibilities to build a model for the (2,3) ISS case where the ALP is the DMcandidate, that is, the Z15−19 symmetries, can be discarded. However, mod-els explaining the DM relic density via ALPs and/or sterile neutrinos forthe (2,3) ISS case can be found, provided we slightly relax some of theconstraints mentioned in section 3.1.

3.2.3.2 ALPs and sterile neutrinos as DM candidates

Actually, if an extra ZN symmetry is allowed, it is found that, for example,the solution (p, |q|) = (2,3) in Eq. (3.17) makes possible a model withD = 10 and (p, q, r, s, t) = (2,−3,−3,2, 2) in Eqs. (3.3), (3.7) and (3.20),where the discrete gauge symmetry is Z10 × Z4, with the correspondingcharges given in table 3, must be considered to obtain the correct DM den-sity. It is straightforward to check that for this model, ALPs provide 100%

3.2 M O D E L S 45

Model ma ×10−11 (eV)gaγγ×10−17

(GeV−1)

M (TeV), µS (keV)

mνlight (eV)

M22a (19.0−56.0) (5.5−7.8)(1.5−4.5), (0.1−0.7),

(1.0−1.4)

M22b (0.52−1.4) (1.1−1.6)(24.3−65.6), (11.3−58.9),

(0.4−0.6)

M33a (19.0−56.0) (2.7−3.9)(1.5−4.5), (0.1−0.7),

(1.0−1.4)

M33b (0.52−1.4) (2.2−3.1)(24.3−65.6), (11.3−58.9),

(0.4−0.6)

M23a (0.06−0.30)×1011 (720−1200)(7.5−21.0), (4.12−19.41),

(1.33−2.24)

M23b (0.03−0.2)×1011 (830−1400)(5.6−15.8), (2.7−12.7),

(1.4−2.5)

TABLE 4: Main features of the models discussed in the text. The constant g ∈[10−3, 2] in the mass term of the ALPs, and the η Yukawa of order 10−2

in the M23a(b) models have been considered. The second and thirdcolumns represent the ALPs parameters (mass and coupling to photons,respectively), while in the last column are given the ISS mass scales.

of the DM abundance when vσ ∼= 2.03× 1011 GeV which is an allowedvalue for D = 10 in table 2. For this benchmark point (a benchmark re-gion denoted as M23a where ALPs provide 100% of the DM abundance isshown in Figure 6) we found that:

gaγγ ' 1.14×10−14

2.03×1011 GeVvσ

GeV−1, (3.47)

ma ' 0.29|g|12

h vσ2.03×1011 GeV

i4eV,

with the neutrino mass spectrum given by:

M ' ζ×8.4 TeV; µS 'h η

10−2

i

×4.96 keV;


O(ζ2)eV. (3.48)

Notice that for η≤ 10−2 and the other coupling constants of order one, asuitable neutrino mass spectrum is achieved. In this case, the sterile neu-trino gives a negligible contribution to DM relic density because the mixingangle between the active and sterile neutrinos is smaller than the limits es-tablished to consider νS as a DM candidate, 10−8 ® sin2(2θ ) ® 10−11,(see ref. [48] for more details).For the case where the DM abundance is made of ALPs and sterile neu-trinos, the scenario slightly changes since it is necessary to consider the


values shown in the last column of table 2. In this work, we have chosenthe case when the DM is made of ≈ 43% of sterile neutrinos and ≈ 57% ofALPs as an illustrative example. However, these can take other values pro-vided the DM abundance made of sterile neutrinos is ¯ 50%, consistentlywith the constraints on its parameter space [48, 236, 245]. Following asimilar procedure as in the previous cases, we obtain, for the particularvalue vσ ∼= 2.28×1011 GeV,

gaγγ ' 1.02×10−14

2.28×1011 GeVvσ

GeV−1, (3.49)

ma ' 0.46|g|12

h vσ2.28×1011 GeV

i4eV,

and the neutrino spectrum

M ' ζ×10.6 TeV; µS 'h η

10−2

i

×7.1 keV;


O(ζ2)eV (3.50)

In this case, for |η| ≈ 10−2 the sterile neutrino has a mass mνS≈ 7.1KeV. In

particular, this mass for the sterile neutrino may explain the emission linesat 3.5 keV from galaxy clusters and the Andromeda galaxy [206, 207]. Thebenchmark region for this model is denoted as M23b in Figure 6.It is worth to mention that for both benchmark regions of these models,the constraints and prospects regarding the LFV processes are similiar tothe ones in the ISS (2,2) case. This happens because the contribution ofthe sterile neutrino to Br(`β → `βγ) is negligible since G(m2

νS/m2

W )→ 0for mνS

mW .Finally, for clearness, in table 4 we give an overview of the main resultsof all models considered. Specifically, we show the scales for the neutrinomasses, and the ALPs parameter space for each ISS case. The results forALPs, reported in Fig. 6, show that it is possible to build a frameworkthat is simultaneously consistent with the phenomenology of neutrinosand ALP, giving an ALPs phenomenology within the reach of next futureexperiments.

4R E S O N A N T P R O D U C T I O N O F D A R K P H O T O N S I NP O S I T R O N B E A M - D U M P E X P E R I M E N T S

In this chapter we will consider the vector portal in order to explain the8Be anomaly present in nuclear transitions through the decay A′→ e+e−.

4.1 I N T R O D U C T I O N

The so far unsuccessful search for new physics BSM has triggered in recentyears an increasing interest in the possibility that the mass scale associatedwith new physic is within experimental reach, but the couplings betweennew particles and the SM are so feeble that a whole new sector has so farremained hidden. This gave rise to many proposals and to many new ideasto hunt for new physics at the intensity frontier [34, 37]. A particularly in-teresting possibility is, the so called DP, a massive gauge boson arising froma new U(1)′ symmetry that can be considered as a natural candidate for asuperweakly coupled new state, since its dominant interaction with the SMsector might arise solely from a mixed kinetic term 1

2εF ′µνFµν coupling theU(1)′ and QED field strength tensors. From the phenomenological pointof view, these light weakly coupled new particles have been invoked to ac-count for discrepancies between SM predictions and experimental results,as for example the measured value of the muon anomalous magnetic mo-ment [50] or the anomaly observed in excited 8Be nuclear decays by theAtomki collaboration [51–53]. This last anomaly is particularly relevantfor the present work since the new experimental technique that we aregoing to describe appears to be particularly well suited to test, at least insome region of the parameter space, the particle physics explanation basedon a new gauge boson with mass mA′ ∼ 17MeV kinetically mixed with thephoton [246].

4.1.1 Beryllium anomaly

The anomaly consists in the observation of a bump in the opening an-gle and invariant mass distributions of electron-positron pairs producedin the decays of an excited 8Be nucleus [51], which seems unaccountableby known physics. The anomaly has a high statistical significance of 6.8σwhich excludes the possibility that it arises as a statistical fluctuation. Theshape of the excess is remarkably consistent with that expected if a newparticle with mass mA′ = 17.0± 0.2(stat)± 0.5(sys) MeV [53] is beingproduced in these decays. The strength of the A′ coupling to e+e− pairs,parametrized as ε =

p

α′/α with α′ the U(1)′ fine structure constant, isconstrained by different experimental considerations. In the Atomki setup,

47

48 R E S O N A N T P R O D U C T I O N O F D A R K P H O T O N S I N P O S I T R O N B E A M -D U M P E X P E R I M E N T S

A′ → e+e− decays must occur in the few cm distance between the target,where the 8Be excited state is formed, and the detectors. This implies alower limit on ε parameter, ε/

Æ

Br(A′→ e+e−) >∼ 1.3× 10−5, (it is leftunderstood that all the limits on ε apply to its absolute value).

4.1.2 DP constraints

In this work, we will assume for simplicity Br(A′ → e+e−) = 1. If theA′ decay with a non-negligible rate into invisible dark particles χ, withmχ < mA′/2, the quoted limits need to be accordingly rescaled. However,in case the invisible decay channel becomes largely dominant, other limitsdifferent from the ones discussed in this work apply (see ref. [247]).Lower limits on ε much stronger than what implied by the Atomki experi-mental setup are obtained from electron beam-dump experiments:

• Old data from KEK [190] and ORSAY [191] have been reanalyzed inref. [180] yielding, in the interesting mass range mA′ ∼ 17 MeV, thelimit ε >∼ 7×10−5;

• A stronger limit, ε >∼ 2×10−4, was obtained in ref. [183] from a re-analysis the E141 experiment at SLAC [136]. However, for a mA′ ∼17MeV the excluded region is very close to the kinematic limit ofthe sensitivity (see Fig. 13) and it has been recently pointed out, bydirect comparison with exact calculations [248], that the Weizsäker-Williams (WW) approximation [249–251] adopted to derive the lim-its become inaccurate in this kinematic region, tending to overesti-mate the reach in mass [248, 252, 253]. In detail, for primary en-ergies in the the range 10− 20 GeV, as was the case for the E141beam [136], and for mA′ ∼ 20MeV, the WW approximation yields anA′ production cross section about 50% larger than the exact calcula-tion, as we can see in Fig. 7, and it also overestimates the A′ emissionspectrum at large energies (see figure 4 in ref. [253]), in which casethe number of expected positrons falling within the 1.1mrad angularacceptance of the experiment would be overestimated both becauseof the larger boost, and also because of the larger lifetime dilationthat would cause the A′ to decay closer to the detector. Besides this,let us note that an A′ slightly heavier than the benchmark value of 17MeV would in any case evade the E141 limit. It is then questionableif, for mA′ >∼ 17MeV, the E141 constraints on the A′ couplings can beconsidered as firmly established. Conservatively, we will assumedthat the corresponding region is still viable;

• Upper bounds on ε in the relevant A′ mass range also exist, see forexample Fig. 13. The KLOE-2 experiment has searched for e+e− →γA′ followed by A′ → e+e− setting the limit ε < 2× 10−3 [185],while constrains from the anomalous magnetic moment of the elec-tron [54] yield ε < 1.4×10−3 [181, 182]. A comparable limit stems

4.2 T H E PA D M E E X P E R I M E N T AT L N F 49

20 40 60 80 100

1.0

1.5

2.0

2.5

E0HGeVL

ΣW

WΣ

EX

AC

T

mA’=100 MeVmA’=50 MeVmA’=20 MeVmA’=10 MeVmA’=5 MeVmA’=1 MeV

FIGURE 7: WW approximation and exact production for A′ in e−Z → e−Z +A′ asa function of the electron energy E0 for different values of A′ masses. Figuretaken from ref. [253].

from BaBar searches for A′ → e+e− decays, but it only applies formA′ > 20MeV [184].

In summary, we will take the interval

7×10−5 ≤ ε≤ 1.4×10−3 , (4.1)

as the window allowed for a 17 MeV A′ decaying dominantly into e+e−.

4.2 T H E PA D M E E X P E R I M E N T AT L N F

We will now discuss some important points related to the DP productionchannels that have been considered for DP experimental searches.

Collider searches for DP have been carried out in electron beam-dump ex-periments (see ref. [180] for a review) assuming A′-strahlung as the lead-ing production mechanism in electron-nucleon scattering. Parametrically,this process is of order α3 as it can be seen in Fig. 8-(a), being α the finestructure constant. As regards A′ searches with positron beams, there areonly few facilities which, in the next future, will be able to provide beamssuitable for fixed target experiments, and correspondingly only a few ex-perimental proposals have been put forth [179, 254, 255]. The produc-tion mechanism considered so far is analogous to the usual QED processof positron annihilation off an atomic target electron with two final statephotons, where one photon is replaced by one A′ as it is shown in Fig. 8-(b), corresponding to a process of O(α2). This is the specific productionprocess envisaged for the Frascati Positron Annihilation into Dark MatterExperiment (PADME) [179] that we will now describe briefly.


A′

γ

γ

e− e−

Z Z

e−

e+

A′A′

e−

e+

(a)

(b) (c)

FIGURE 8: A′ production modes in fixed target e−/e+ beam experiments. (a) A′-strahlung in e−-nucleon scattering; (b) A′-strahlung in e+e− annihilation; (c) res-onant A′ production in e+e− annihilation.

4.2.1 Experimental setup

The PADME experiment [179, 256] at the DAΦNE LINAC Beam Test Fa-cility (BTF) [257] of the INFN Laboratori Nazionali di Frascati (LNF) hasbeen designed to search for DP by using a positron beam [189] imping-ing on a thin target of low atomic number. The A′ can be detected inthe invisible channel by searching for a narrow bump in the spectrum ofthe missing mass measured in single photon final states, originated viathe process e+e− → γA′. The experiment will use a 550 MeV positronbeam impinging on a 100µm thick active target made of polycrystallinediamond (Z = 6). To keep under control the counting rates the beam in-tensity will be kept at ∼ 1013 positrons on target per year (pot/yr), thatis well below the maximum available intensity (see table 5). The low Zand the very thin target are intended to minimize the probability of pho-ton interaction inside the target since, in order to reconstruct accuratelythe missing mass, the measurement requires a precise determination ofthe four-momentum of the γ produced in the annihilation. The recoil pho-tons will be detected by a quasi cylindrical calorimeter made of inorganiccrystals located 3.3 m downstream the target, while the non-interactedpositrons, which constitute the vast majority of the incoming particles, aredeflected outside the acceptance of the calorimeter by a 1 m long dipolemagnet. Three different sets of plastic scintillator bars will serve to detectelectrons and positrons (see Fig. 9). Profiting by the presence of a strongmagnetic field, these detectors, intended to provide an efficient veto forthe positron bremsstrahlung background, can also be used to measure the

4.3 A′ P R O D U C T I O N V I A R E S O N A N T e+ e− A N N I H I L AT I O N 51

FIGURE 9: Schematic of the PADME Experiment. Figure taken from Ref. [258].

charged particles momentum (for more detail see Refs. [179, 258]). ThePADME detector is thus able to detect photons and charged particles andit will be sensitive to invisible (A′→ χχ) as well as to visible (A′→ e+e−)DP decays.

4.3 A′ P R O D U C T I O N V I A R E S O N A N T e+ e− A N N I H I L AT I O N

In this work we point out that for A′ masses ¦ 1 MeV, the process ofresonant e+ e− annihilation into on-shell A′ depicted in Fig. 8-(c), rep-resents another production mechanism which, being of O(α), is paramet-rically enhanced with respect to the previous two production channels.Besides this, A′ production via resonant e+ e− annihilation has severalother advantages that we will illustrate below, which altogether suggestthat it might be particularly convenient to operate the PADME (as well asother) positron beam fixed target experiment in order to search for A′ viaresonant production. Besides experiments with positron beams, resonante+ e− → A′ annihilation must also be accounted for in a correct analysisof electron beam-dump experiments since, as is remarked in ref. [60, 61],positrons are abundantly produced in the electromagnetic (EM) showersinside the dump. This feature was recently exploited in ref. [60] in re-analysing old results from the SLAC E137 experiment [137] by includingA′ production via resonant annihilation (and, but less importantly, alsoA′-strahlung in annihilation). As a result, it was found that due to the con-tribution of resonant A′ production, the E137 data exclude a parameterspace region larger than it was previously though [180, 183] (see discus-sion “electromagnetic shower” at the end of this chapter). The extendedexcluded region corresponds to the area in light grey color towards the bot-tom of the plot in Fig. 13. Hence, in analysing electron beam-dump data,A′ production from annihilation of secondary positrons via the diagramsin Fig. 8-(b) and (c) should be also accounted for.


pot/yr E min (MeV) E max (MeV)

e+ 1018 250 550

e− 1018 250 800

TABLE 5: Beam parameters for the Frascati BTF. The second column correspondsto the number of positrons on target per year, while the last two columnsprovide the energy limits of the beam.

Taking into account the importance of resonant annihilation to create DP,we will consider the sensitivity of the PADME experiment to the productionprocess e+ e− → A′ → e+ e−. In order to exploit this resonant productionmechanism, however, an experimental setup slightly different from the oneoriginally conceived is more convenient. The thin diamond target shouldbe replaced by a tungsten target of several cms of length, and this for twomain reasons:

1. Several cms of tungsten can absorb most of the incoming positronbeam and of the related EM showers, and in any case to degradesufficiently the energy of the residual emerging particles, so that thecharged particles can be easily deflected and disposed of. The A′ pro-duced in e+e− annihilation, if sufficiently long lived, will escape thedump without interacting, and will decay inside the downstreamvacuum vessel, producing an e+e− pair of well defined energy. Thethick tungsten target thus allows to take advantage of the full beamintensity of 1018 pot/yr, with a gain of five orders of magnitude withrespect to the thin target running mode (see table 5). 1

2. A thick target can provide an almost continuous energy loss for theincoming positrons propagating through the dump, so that they canefficiently “scan” in energy for locating very narrow resonances.

To correctly consider this second point it is necessary to study the en-ergy distribution of positrons inside the BTF beam, which is tunable toa nominal energy Eb within the range 250 ≤ Eb

MeV ≤ 550. This distribu-tion can be described by a Gaussian function G(E) = G(E; Eb,σb) whereσb/Eb ∼ 1% is the energy spread. On the other hand, the probabilitythat a positron with initial energy E will have an energy Ee after travers-ing t = ρ · z/X0 radiation lengths (with ρ the density of the material ing/cm−3 and X0 = 6.76 g/cm−2 the unit radiation length in tungsten), isgiven by [259, 260]:

I(E, Ee, t) =θ (E − Ee)

E Γ ( 4t3 )

log

EEe

4t3 −1

, (4.2)

1 The maximum number of e± deliverable in one year given in the table is LNF site autho-rization limited by the efficiency of the existing radiation shielding. However, technicallythe BTF could deliver up to 1020 electrons or positrons on target per year.


where Γ is the gamma function. Eq. (4.2) neglects secondary positronsfrom EM showers, as well as the loss of primary positrons from e+e−→ γγannihilation, but is still sufficiently accurate for our purposes. Taking intoaccount the effects described for the function in Eq. (4.2), the e+ energydistribution after t radiation length is given by:

T (Ee, t) =

∫ ∞

0

G(E) I(E, Ee, t) dE . (4.3)

Integrating T (Ee, t) in t it is possible to obtain the track-length distributionfor the primary positrons. However, for an accurate determination of thedetectable number of A′, the coordinate z = tX0/ρ of the production pointis important, especially for the larger ε values, and hence shorter decaylengths since the A′ decay width is ΓA′ '

13ε

2αmA′ . Thus, the integrationin t should be performed only when accounting for the probability of A′

decaying outside the dump. To taking into account this we fix the origin ofthe longitudinal coordinate at the beginning of the dump, such that zD isthe end point of the dump and zdet is the distance between the origin andthe detector.Note that the A′ decay length,

`ε = c γτA′ , (4.4)

with γ= mA′2me

the time dilation factor, depends quadratically on ε through

the lifetime τA′ =1ΓA′

but it does not depend on mA′ . Therefore, for the

range of ε given in Eq. (4.1) we find that 16>∼`εmm>∼ 0.04.

With the previous considerations, the number of detectable DP events canbe estimated as:

NA′ =Ne+N0X0Z

Ae−

zD`ε

∫ T

0

d t eX0ρ`ε

t∫ ∞

0

dEe T (Ee, t)σ res(Ee) , (4.5)

with Ne+ the number of incident positrons, N0 the Avogadro number, A=

184 the atomic mass of tungsten, Z = 74 the atomic number and σ res(Ee)

the differential resonant cross section of e+e−→ A′→ e+e− defined below.Eq. (4.5) takes into account the fact that the probability to detect an A′

produced at z is given by the integral of

dPdz

=1`ε

e−z`ε (4.6)

between zD− z and zdet→∞, where the limit is justified since zD ∼O(1)m. Moreover, if the initial beam energy happens to be not much abovethe resonance, after just a fraction of a radiation length (X0/ρ = 3.5mmfor tungsten) the energy of most positrons will have already degraded be-low the threshold for resonant production, so that setting T = 1 for theupper limit of the integration is also a good approximation. We can seeschematically this behavior in Fig. 10.In Eq. (4.5) the first exponential accounts for the fact that the larger is thelength of the dump, the smallest is the number of A′ that can be detected.


Eres

t=0

t=0.05

t=0.10

t=0.25

t=0.50

275 280 285 2900.0

0.2

0.4

0.6

0.8

1.0

Eo(e+) (MeV)

FIGURE 10: Evolution of the positron energy distribution at different radiationlengths t. The vertial axis represent the positron energy distribution T (Ee, t). Allthe curves are normalized to one. The vertical narrow black strip represents theA′ resonance of a 17 MeV DP.

For zD ∼ 10 cm we can expect that virtually all the background from theEM showers will be absorbed in the dump. However, only a few A′ will de-cay outside. To increase the statistics we can reduce zD, but then keepingthe background under control can become an issue. In the lack of a dedi-cated simulation of the detector/background for the resonant annihilationprocess, we will estimate the sensitivity to the A′ couplings that could beachieved with zD = 10cm, zD = 5 cm, and zD = 2cm (in the last twocases a reduction of the beam intensity to keep under control backgroundcontamination might be required).As regards σ res, the resonant s-channel amplitude for e+e−→ A′→ e+e−

does not interfere with the analogous QED process with an off-shell γ, norwith t-channel amplitudes that can then be neglected. Using the narrowwidth approximation σ res can be written as:

σres(Ee) = σpeakΓ 2

A′/4

(p

s−mA′)2 + Γ 2A′/4

, (4.7)

with σpeak '12πm2

A′and s ' 2meEe, the e+e− system invariant mass squared.

It is important to comment that in the numerical computation we take intoaccount me effects both in the cross section and in the width, and we alsoaccount for the emission of soft photons from the initial state (see e.g. ref.[261] and appendix B) up to energies∆E/Eb ≈ 1%, which can radiativelyenhance the resonance width, and thus the production rate.

Important comments

With respect to other DP production mechanisms, resonant production hassome peculiarities and advantages:


279 280 281 282 283 284 285Ee (MeV)

10 1

100

101

102

103

104

105

106

Num

ber

ofA′

Eres = 282.3 MeV= 10 4

zD = 10 cm

FIGURE 11: The number of DP decaying outside the dump as a function of thebeam energy for ε = 10−4. The vertical line corresponds to the energy for res-onant production of a 17 MeV DP. A dump length zD = 10cm and a backgroundfree measurement have been assumed.

(i) The peak cross section does not depend on ε and the dependence ofthe total resonant cross section is only quadratic (∼ ε2α). As regardsthe observable number of electron-positron pairs from A′ decays, forsmall ε the suppression in production is over-compensated by thestrong enhancement from the larger decay length∼ exp(−ε2) whichincreases the number of A′ that decay outside the dump (check Eq.(4.5)). For this reason, resonant DP production in thick target exper-iments is particularly well suited to explore the parameter space atsmall ε.

(ii) At fixed value of ε, the A′ decay length `ε = γ cτA′ is independentof the value of the A′ mass. This is because mA′ cancels between theboost factor γ ∼ mA′

(2me)and the lifetime τA′ ∝

1mA′

. For all A′ masses

the decay length is then fixed to be `ε ∼3

(2meαε2) . Therefore, theentire mA′ range within the reach of the beam energy can be probedwith the same sensitivity.

(iii) Under the reasonable assumption that the background remains con-stant when the beam energy is varied by only a few MeV, the back-ground can be directly measured from the data. This is illustratedin Fig. 11 where we can see that when the beam energy lies wellbelow the resonance, the background for e+e− pairs (assumed to beabsent for the case of zD = 10 cm show in the picture) can be directlymeasured. When the beam energy is increased, in approaching res-onant production the number of e+e− pairs produced increases in astep-wise way up to a maximum, and then remains approximatelyconstant with increasing energy, due to positron energy losses in thematerial, which drive their energy towards E res. Clearly, even in thepresence of a significant number NBG of e+e− background pairs, aslong as NA′ >

p

NBG a signal of A′ decays can be detected.


Such a spectacular signature would be prevented if the A′ resonancelies somewhat below the minimum beam energy, since one would al-ways measure e+e− resonantly produced by primary e+ degraded inenergy, together with backgrounds. However, in this case by raisingthe beam energy and stepping further away from the resonance, thenumber of dilepton pairs resonantly produced would drop becauseof the degradation of the primary beam quality due to EM shower-ing. The behavior of a “background” which decreases with increasingbeam energy would still be a signal of beyond the SM physics.

4.3.1 Effects of target electrons velocities

So far all the considerations about the beam-dump experiment take theelectrons inside the target as at rest, but actually, these are not at rest,and in the case of large atomic numbers, like tungsten 74W, the electronscan have large velocities, especially the ones in the inner core shells. This

can be verified by estimating the electrons virial velocities ⟨vnl⟩ ≈ αZ (nl)eff in

terms of the effective nuclear charge Z (nl)eff felt by electrons in the (nl) shell

(a complete list of effective nuclear charges can be found in ref. [262]).For targets of small atomic number, like 6C or 13Al, virial velocities aresmall, and the effects of target electrons motion is likely to be negligible.However, for 74W one finds that the average velocities span a rather largerange 0.003<∼ ⟨vnl⟩<∼ 0.5 when going from valence or conduction elec-trons (with Fermi energy εF ∼ 4.5 eV) to inner core electrons. Thus, forpositron annihilation in tungsten the center of mass (c.m.) energy can dif-fer sizeably from what can be naively estimated in terms of the beam en-ergy, energy spread, energy loss due to in-matter propagation, and assum-ing electrons at rest. To give an example, already for a longitudinal velocitycomponent vz ∼ 0.03 the effect of shifting the c.m. energy away from theresonant value is three time larger than the effect of the intrinsic ∼ 1%energy spread in the beam energy. Of course, what is needed to accountfor the c.m. energy shift is not simply the momentum distribution of elec-trons, but rather the positron annihilation probability as a function of theelectron momentum, since the annihilation with delocalized and weaklybound valence electrons, which contribute to the low-momentum part ofthe momentum distribution, is more likely than annihilation with the local-ized and tightly bound core electrons contributing to the high-momentumpart.For positron annihilation at rest, the annihilation probability distributionas a function of the electron momentum is directly measured from theDoppler broadening by the amount ∆E =

pL2 of the 511 keV photon line,

with pL the e− momentum component along the direction of γ emission(the relative direction of the two γ’s also deviates from 180o by the smallangle θ =

pLme

). In Fig. 12 (adapted from ref. [263]) a large set of exper-imental points for 74W is represented with blue crosses. The red dashedline represents a theoretical calculation performed in the same paper. Up to


0 10 20 30 40 50 60

pe − (10−3me)

10-6

10-5

10-4

10-3

10-2

10-1

100

P(pe−)

Experimental

Calculated

Fitted

FIGURE 12: Positron annihilation probability as a function of the target electronmomentum for tungsten (figure adapted from ref. [263]). The blue crosses repre-sent experimental points, while the red dashed line is the result of the calculationmethod adopted inref. [263]. The green dot-dashed line corresponds to the fitgiven by the function in Eq. (4.8).

pe− ∼ 15×10−3 me the main contribution to the annihilation comes fromelectrons in the 5d shell, beyond that point 4f electrons dominate, whilethe contribution of the high momentum core electrons becomes relevantonly for pe− >∼ 40× 10−3 me where, however, the annihilation probabilityis suppressed below 10−5. Accordingly, we find that a good fit to the exper-imental and calculated distributions [263] can be obtained with the sumof just three terms:

P(ve) =112

1.015−v2e + 1.112−2ve + θ (ve −40)3×10−6+ 1

ve

, (4.8)

where ve =pe−me

. The first term in parenthesis accounts for 5d electrons, thesecond for 4f electrons, and the last one, which is non zero only for ve ≥ 40,accounts for core electrons. To take into account the target electron motionwe thus replace the Mandelstam variable s in the expression for σres by:

s(ve,χ) = 2me

Ee

1−P(ve)ve12

sχ cχ

+me

, (4.9)

where cχ = cosχ accounts for the projection of ~ve along the z-directionof the incoming positron, and, 1

2 sχ , with sχ = sinχ, is the probabilitydistribution for the angle χ. After this replacement we integrate the crosssection in cχ and ve ∈ [0, 0.06].Table 6 collects some results that illustrate how the number of DP producedwithin the first radiation length of tungsten depends on various effects. Thesecond column gives the results for three different values of ε for a beamenergy tuned at the resonant energy E res = 282.3MeV, when the motionof the target electrons is neglected. The third column gives the resultsobtained when the electron velocity is taken into account according tothe distribution in Eq. (4.8). Note that the shift of the c.m. energy due


ε.

N prodA′ E res (ve = 0) E res E res + 2σb

1.0×10−3 7.69×1011 1.51×1011 4.72×1011

5.0×10−4 1.81×1011 3.79×1010 1.17×1011

1.0×10−4 7.25×109 1.49×109 4.73×109

TABLE 6: Number of 17MeV DP produced in the first radiation length of a tung-sten target for 1018 positrons on target, for three different values of ε.The second and third columns are for a beam energy tuned to the reso-nant value Eres = 282.3 MeV, assuming respectively electron at rest andwith the velocity distribution in Eq. (4.8). The last column, also includ-ing ve effects, is for a beam energy tuned to Eb = E res + 2σb, whereσb/Eb ∼ 1% is the energy spread.

to the electron momentum has the effect of reducing the number of DPproduced by about a factor of five. The last column gives the results fora beam energy tuned above the resonance Eb = E res + 2σb. In this case,the number of DP is increased by about a factor of three because of thepositron energy losses, which brings on resonance also positrons in thehigh energy tail of the initial energy distribution.An important comment to consider is that using the annihilation probabil-ity distribution for positrons at rest in the problem at hand, can be a crudeway of proceeding. We can expect that target electron motion effects canbe more sizeable for in-flight annihilation of short wavelength positronswith energies of O(100) MeV, since the annihilation probability with elec-trons in the inner shells will be enhanced. Therefore, our estimate for theproduction rates might be optimistic by a factor of a few. On the other hand,while positron energy loss, which proceed mainly via bremsstrahlung, con-stitute a quantized process, the dependence of the c.m. energy on the angleχ characterizing the electron momentum is continuous, and this justifiesmodeling positron energy losses as a continuous process.

4.4 R E S U LT S A N D D I S C U S S I O N S

In this section, we will discuss the main results in the implementation ofA′ resonant production. Some important comments about the backgroundare also given.

Comments about the background

• The PADME spectrometers can detect e+e− pairs with good resolu-tion for coincidence in time and momentum. The A′ angular spreaddue to the transverse momentum of atomic electrons is much lessthan the intrinsic angular spread of the beam (∼ 1 mrad) and itdoes not affect the reconstruction of the coincidence.

4.4 R E S U LT S A N D D I S C U S S I O N S 59

• For targets of sufficient thickness, background from secondary e− de-tected in coincidence with primary or secondary e+ can be avoidedby measuring their depleted momentum via electromagnetic deflec-tion. Meanwhile, for targets of smaller length a certain number ofe+e− pairs retaining a large fraction of the beam energy can exit thedump, and in this case the data driven method of searching for a“knee” in the number of e+e− pairs versus beam energy (see Fig. 11)can provide a precious tool for revealing the onset of resonant e+e−

production on top of the background.

• Punch-through photons, produced via bremsstrahlung in the veryfirst layers of the dump, carrying a large fraction of the original beamenergy, and converting in e+e− in the last millimeter or so, constitutethe most dangerous background. This background could be signifi-cantly suppressed by equipping the experiment with a plastic scin-tillator veto few mm thick, or a silicon detector of a few hundredsof µm, placed right at the end of the dump, to ensure that the e+e−

pairs originate from decays in the vacuum vessel outside the dump.Additionally, if the experiment could be equipped with a suitabletracker, able to provide an accurate e+e− invariant mass reconstruc-tion, many sources of backgrounds could be further reduced. In par-ticular, given that the invariant mass of the e+e− originating fromphoton conversion m2

e+e− = 0 is very far from m2e+e− ∼(17 MeV)2 ex-

pected from resonant annihilation, the punch-through photon back-ground could be efficiently eliminated.

In Fig. 13 we shown the status of the current limits for DP searches as-suming visible A′ decays into e+e− pairs with unit branching fraction andsuppressed couplings to the proton. As is discussed in ref. [55] the last as-sumption is required in order to evade the tight constraints from π0→ γA′

obtained by the NA48/2 experiment [264], and to render thus viable anexplanation of the 8Be anomaly via an intermediate A′ vector boson. Forthis reason we have not included in Fig. 13 the limits from the NA48/2experiment [264] nor those from the ν-Cal I experiment at the U70 accel-erator at IHEP Serpukhov [265, 266] which also do not apply for proto-phobic A′. In the figure, the vertical black line gives the location of the DPresonance at mA′ = 17 MeV. Leaving aside the limits from the SLAC E141 The two PADME

search modes arecomplementary,since they can setnew boundsrespectively in theregions of largeO(10−3) and smallO(10−4) values ofthe DP mixingparameter ε.

experiment for which the reach in A′ mass might be overestimated, a vi-able window remains between the Orsay/KEK lines (ε >∼ 7×10−5) and the(g − 2)e line (ε <∼ 1.4× 10−3). In the same plot, the black hatched regiondepicts the forecasted sensitivity of PADME in thin target mode, that willsearch for DP via the e+e− → A′γ process. The limits assume 1013 pot/yr.The light cyan trapezoidal regions represent instead the constraints thatPADME could set by running in thick target mode with 1018 pot/yr, andare respectively for tungsten targets of 10 cm, 5 cm and 2 cm of length(neglecting backgrounds).


10-2 10-1

mA ′ (GeV)

10-8

10-7

10-6

10-5

10-4

10-3

10-2

ε

BABAR

(g - 2)e

E137KEK

E774

Orsay

KLOEPADME (e + e − →A ′γ)

E141zD=2 cmzD=5 cm (e + e − →A ′)reszD=10 cm

FIGURE 13: Limits on the DP kinetic mixing ε as a function of the mass mA′ fromdifferent experiments [59]. The region that could be excluded by PADME runningin thin target mode is hatched in black, while the three trapezoidal-shaped areasgive the PADME reach in thick target mode, respectively for a 10, 5 and 2 cm tung-sten dump, assuming zero background. The lower region in light gray extendingthe E137 exclusion limits is from the reanalysis in ref. [60].

The BTF energy range for positron beams 250<∼Eb

MeV<∼ 550 corresponds

to c.m. energies in the interval 16<∼E c.m.MeV

<∼ 23.7. Neglecting a possiblesmall c.m. energy increase from target electron velocities, the upper valuesets the upper limit on the A′ masses that can be produced. The lowerc.m. energy limit is indicated by the thin vertical brown line inside thetrapezoidal regions. However, because of positron energy losses, the regionat low mA′ that can be explored extends to values smaller than 16 MeV, asindicated in the figure. Of course, in propagating well inside the dump, thebeam gets degraded in energy, directions of particle momenta and numberof positrons, by several effects that we are neglecting. Therefore, we canexpect that the experimental sensitivity could be extended down to mA′

values lower than 16 MeV by no more than a few MeV. This might still besufficient to reach into the region where the E141 exclusion limits can betrusted.Before concluding, we will discuss some important remarks. A new way tosearch for narrow resonances, and specifically DP, coupled to e+e− pairs,via resonant production in e+e− annihilation is suggested. In this way animportant region of the parameter space can be tested by the PADME ex-periment (see Fig. 13). However, we can see that there is a gap betweenthe large ε region that can be bounded by searching for A′ produced viae+e−→ γA′ (hatched black region), and the small ε region that can be ef-ficiently explored via resonant e+e−→ A′ production (trapezoidal-shapedareas). The reason for this gap is that the first process, being of O(α2ε2),looses quickly sensitivity when the value of ε is decreased too much, whileA′ production via resonant annihilation becomes inefficient when ε be-


comes too large, so that most A′ → e+e− decays occur inside the dump.Resonant e+e−→ A′ production is not relevant for PADME running in thintarget mode, because the large beam energy (Eb ∼ 550MeV) implies thatpositrons will always have energies far from any narrow resonance withmass smaller than 23.7 MeV, given that positron energy losses in the thindiamond target (∼ 100µm) are negligible. 2

To finish this chapter, we will show the main results obtained in ref. [60,61]. These works exploit the A′ resonanta production mode considering asubsequent decay into visible and/or invisible sector. In those, accelerator-based experiments that make use of an intense electron beam of moder-ate energy (∼ 10− 20 GeV) dumped on a thick target are sensitive to awide area of the DP parameter space. In this case, high-energy positronannihilation plays a significant role since the positrons produced by theelectromagnetic shower can produce an A′ via nonresonant and resonantannihilation on atomic electrons.

Electromagnetic shower

When a high-energy electron impinges on a material, it initiates an elec-tromagnetic shower, that is, a cascade multiparticle production process.The two main reactions contributing to the process are photon produc-tion through bremsstrahlung by electrons and positrons and e+e− pair pro-duction by photons. As a consequence, after a few radiation lengths, thedeveloping shower is made by an admixture of electrons, positrons, andphotons, characterized by different energy distributions. In previous pa-pers describing A′ production in electron beam-dump experiments [180],only the bremsstrahlung like DP production by electrons has been included.Compared to the characteristic shape of A′-strahlung exclusion limits, theresonant annihilation process leads to a more stringent constraint at lowε, in the A′ mass window constrained by the primary beam energy (rightbound) and by the detection threshold (left bound).

For visible decays, the contribution of resonant annihilation results in alarger sensitivity with respect to limits derived by the commonly usedA′-strahlung in certain kinematic regions. In particular, when is includedin the evaluation of the E137 beam-dump experiment reach [137], thepositron annihilation pushes the current limit on ε downwards by a factorof 2 in the range 33 MeV< mA′ < 120 MeV. The E137 experiment is theone sensitive to the smallest values of ε, down to ∼ 10−8 (see Fig. 13).It searched for long-lived neutral objects produced in the electromagneticshower initiated by ∼20 GeV electrons in the beam dump. Particles pro-

2 It is conceivable that by reducing the beam energy down to ∼ 282MeV, by increasing thesize of the target to several 100µm to enhance A′ resonant production, and keeping thebeam intensity well below 1018pot/yr to keep counting rates inside the detector undercontrol, at least part of the remaining region for the 17MeV DP could be explored, andmaybe the whole gap could be closed.


duced in the water-cooled aluminum plates forming the dump would haveto penetrate ∼179 m of earth shielding and decay in the ∼204 m regiondownstream of the shield. The original data analysis searched for ALPs de-caying in e+e− pairs, requiring a deposited energy in the calorimeter largerthan 1 GeV with a track pointing to the beam-dump production vertex.The absence of any signal provided stringent limits on axions and photi-nos. Negative results were used in ref. [180, 248] to set strong constraintson the visible decay A′ → e+e− assuming the A′-strahlung (dark gray re-gion in Fig. 13) as the only production mechanism. Including the resonantand nonresonant positron annihilation, we have derived extended a moreaccurate limits for the A′ coupling to SM particles (lower region in lightgray in Fig. 13).In a general framework (for a more detailed description we invite thereader to see discussions in ref. [60]), to evaluate the A′ production bypositrons and the subsequent detection of the visible decay products (e+e−

pairs) in an electron-beam dump experiment is employed a specific MonteCarlo procedure. In a first step, the energy spectrum and the multiplicityof secondary positrons in the beam dump are evaluated. Then, the resultsare used as input for a custom Monte Carlo code that generates A′ eventsaccording to the two positron annihilation processes, handling the A′ prop-agation and subsequent decay to an e+e− pair, and including the experi-mental acceptance of a detector placed downstream of the dump (the codealso computes the total number of produced particles per electron on tar-get). Decoupling the A′ production in the dump from the electromagneticshower development allows handling cases with ε 1.To derive the E137 exclusion limits for resonant and nonresonant A′, theMonte Carlo based numerical approach described above has been used.The experimental acceptance was evaluated for the E137 runs. In the cal-culation, was employed the same selection cuts used in the original anal-ysis: (i) The energy of the impinging e+e− particle has to be larger than 1GeV. In the case of resonant production, this puts a hard limit on the mini-mum value of the A′ mass of about 33 MeV. (ii) The angle of the impingingparticle on the detector, measured with respect to the primary beam axis,has to be smaller than 30 mrad. Based on the null observation reportedby E137, we derived the exclusion contour considering a 95% C.L. upperlimit of three events. Fig. 14 shows results for both resonant (short-dashedblue line) and nonresonant (long-dashed red line) annihilation. Limits ob-tained by the A′-strahlung from ref. [180, 248] are shown in the figureas a black solid line and a black dotted line, respectively. Resonant anni-hilation provides the best exclusion limits for mA′ in the 33 MeV< mA′ <

120 MeV range, strengthening by almost a factor of 2 the previous limits.The lowest limit on ε ∼ 10−8 is obtained for mA′ =33 MeV. In the case ofresonant annihilation, the sharp cutoff at a low mass is determined by theenergy detection threshold. At large ε, the reach is limited by the small A′

decay width, not sufficiently dilated by the Lorentz boost factor and thusresulting in the A′ decay within the shielding. The nonresonant contribu-


) 2 (MeV/cA’M10

210

ε

8−10

7−10

6−10

5−10

4−10

γ A’ →

e+e

A’→

e+e

A’ strahlung

FIGURE 14: Exclusion limits on the DP parameter space derived from the E137 ex-periment considering nonresonant (long-dashed red line) and resonant (short-dashed blue line) production. Figure taken from ref. [60].

tion is slightly less sensitive but extends the reach to lower masses downto mA′ ∼few MeV, for ε values ranging from ∼ 2×10−8 to ∼ 4×10−8 withrespect to the limit obtained by considering the A′-strahlung.In the case of invisible decay, the idea of positron-rich environment pro-duced by the electromagnetic shower is applied to beam-dump experi-ments and active beam-dump experiments in the work developed in ref.[61] searching for light DM through A′ invisible decay. The results pushdown the current limits by an order of magnitude. Specifically, the contri-bution of positron annihilation for past and future electron beam-dumpexperiments is calculated3 using the Monte Carlo simulation describedabove.In the scenario of light DM with a mass in the range∼(1 MeV – 1 GeV) andcharged under a new U(1)D broken symmetry, the DP can be kineticallymixed with the SM hypercharge field allowing an SM-DM interaction. Inthe particular case where the mass of the DM candidate is smaller thanmA′/2, the A′ decaying into invisible sector (light DM) is a dominant sce-nario. This picture is compatible with the well-motivated hypothesis of aDM thermal origin, where the present DM density is a relic remnant ofits primordial abundance. This hypothesis provides a relation between theobserved DM density and the model parameters.In beam-dump experiments, besides the cascade of SM particles, electronsor positrons stopped in the beam dump may produce an A′ decaying toa χ/χ particle pair, resulting in an effective light DM secondary beam.Having a small coupling to ordinary matter, light DM particles propagatethrough the shielding region to the detector. Scattering on electrons andnuclei of the detector active material may result in a detectable signal. On

3 The sensitivity of the same experimental setups replacing the e− beam with an e+ beamis also investigated. See ref. [61] for a more detailed description.


)2 (GeV/cχm2−10 1−10

y

15−10

14−10

13−10

12−10

11−10

10−10

9−10

8−10

7−10

6−10

I

II

III

BaBar

LSND

E137

BDX

LDMX

NA64

FIGURE 15: Exclusion limits in electron beam-dump experiments. For electron ac-tive beam-dump experiments and beam-dump experiments due to resonant andnonresonant positron annihilation. Dashed lines show exclusion limits obtainedby considering A′-strahlung only. The combined exclusion region is shown as afilled area. Light gray indicates previous limits [267, 268]. Dark gray shows theeffect of including positron annihilation on existing limits. Different colors corre-spond to the different experiments. Figure taken from ref. [61].

the other hand, active beam-dump experiments, use the active dump asa detector, exploiting the missing-energy signature of produced and unde-tected χ to identify the signal. The active dump, detecting the EM shower,allows to measure the energy of individual leptons of a monochromaticbeam, provided a beam current is low enough to avoid pile-up effects.When an A′ is produced, its invisible decay products will carry away asignificant fraction of the primary beam energy, thus resulting in a visi-ble defect in the energy deposited in the active dump. Signal events areidentified when the missing energy exceeds a minimum value.In the approach developed in ref. [61], the new exclusion limits at 90% C.L.obtained considering the positron annihilation mechanisms are shown inFig. 15. In this, limits derived by including A′-strahlung only (dashed lines),positron annihilation only (continuous lines), and the combination of thetwo on existing limits (filled area) are shown. The light-gray area showsthe excluded region before this work, in the parameter space defined byy ≡ αDε

2(mχ/mA′)4 vs mχ , assuming αD = 0.5 and mA′ = 3mχ . The

dark-gray area highlights the contribution of the positron annihilation tothe previously excluded area. The three continuous black lines representthe thermal relic target for different hypotheses on the light DM nature:elastic and inelastic scalar (I), Majorana fermion (II), and pseudo-Diracfermion (III). For some selected kinematics, positron annihilation pushesdown by an order of magnitude the exclusion limits. The shape of activebeam-dump experiments lines is related to the high missing-energy thresh-old. For this class of experiments, the sensitivity at low masses is strongly


connected to the threshold value, resulting in a sharp dip. For beam-dumpexperiments, instead, the threshold effect is less pronounced. Here the en-ergy threshold is usually lower, and the dependence on threshold of thesensitivity for low masses is weaker, resulting in a wider and smoothershape. These results show that positron annihilation needs to be includedfor a correct evaluation of all the light DM exclusion limits obtained fromelectron beam-dump experiments.

5C O N C L U S I O N S

Searches for new light, weakly coupled particles are motivated by someof the most important questions in particle physics as DM, the strong C Pproblem and some discrepancies between the SM and experimental results.The searches are dependent on the tools and techniques of the intensityfrontier, i.e., intense beams of charged particles (protons, electrons andpositrons), and on extremely sensitive detection techniques. Besides thenew light particles may constitute the DM, or be the force carriers respon-sible for its interactions, they may even help explain the origin of astrophys-ical anomalies, or could be low-energy remnants of physics at the highestenergy scales.

In this work, the more important details of two portal connection betweenthe DM and the SM of particle physics have been studied. We exploredsome particles that either couple directly to the photon or mix with it:axions/ALPs and dark photons residing in hidden-sectors. In a first case,we have shown that both puzzles of neutrino masses and of the relic DMmay be related to each other through a model containing ALPs and newphysics at the TeV scale. Specifically, we proposed a model based on theISS mechanism considering the minimal versions in agreement with allthe neutrino constraints. In this framework, the mass scales for the ISSmechanism are generated by gravity-induced non-renormalizable opera-tors when the scalar field containing the ALPs gets a vev vσ. Naturalnessof these scales imposes strong constraints on these operators and, whencombining these with the ALPs acceptable range for vσ, only two solutionsare possible. This implies that operators given M and µS scales can onlybelong to these two categories. Then, a simultaneous application of con-straints coming from the texture of ISS mass matrix, the violation of theunitarity, the mass of exotic charged leptons, the stability of the effectiveLagrangian against gravitational effects and the suitable ALPs parameterspace (ma and gaγγ) to provide the total DM density, almost set the restof terms in the Lagrangian, leaving only a few of possibilities for all ISScases. These constraints ultimately lead to a concrete prediction for theviable ALPs masses and ALPs-photon couplings and also for the mass scaleof the heavy neutrinos necessary to explain the neutrino oscillation data.In other words, both sectors are deeply connected and the observation ofa hypothetical signal of the ALP within the proper regions will automati-cally imply the existence of heavy neutrino states in the TeV and multi TeVrange.Among the minimal ISS mechanisms analyzed, the (2,2) and (3, 3) ISScases are quite similar. It is due to the fact that in both of them nNR

= nSR

67

68 C O N C L U S I O N S

implying that neutrino mass spectrum is characterized by only two massscales, M and mνlight. Thus, the results obtained are almost identical. How-ever, there is a slightly difference in the value of gaγγ due to the presenceof more fermions in the (3,3) ISS case. In both cases, we find two effectivemodels denoted as M22a,b and M33a,b listed in table 4. Since there is nosterile neutrino in these cases, the total DM density is made of ALPs.On the other hand, in the (2, 3) ISS case, the presence of a sterile neutrinoin the mass spectrum implies that the DM density can be made of ALPs andsterile neutrinos. We have found a model satisfying all the previously men-tioned constraints and, at the same time, producing the total DM. Becausesterile neutrinos in the (2,3) ISS mechanism can give, roughly speaking, atmost≈ 43% of the DM density, it is necessary that the remaining≈ 57% ofDM be made of ALPs. It is also possible that ALPs give the total DM density.This occurs when the mixing angle between active and sterile neutrinosis very suppressed in order to make the the Dodelson-Widrow mechanisminefficient. Both cases were studied in detail and denoted as M23a andM23b, respectively.Regarding the search for ALPs, the benchmark regions in Fig. 6 are out ofreach of the current and future experimental searches for axion/ALPs suchas ALPS II, IAXO, CAST [141], since these currently have not enough sensi-tivity to probe the ALPs/axion-photon couplings and masses that are moti-vated in models with scales vσ ¦ 1013 GeV. Nevertheless, for the (2,2) and(3,3) ISS cases the benchmark regions are remarkably within the targetregions in proposed experiments based on LC circuits [156, 213], whichare designed to search for QCD axions and ALPs and cover many ordersof magnitude in the parameter space, well beyond the current astrophysi-cal and laboratory limits [102, 140, 141]. Specifically, the ABRACADABRAexperiment [156] may explore ALPs masses as low as ∼ 10−10 eV for acoupling to photons of the order of ∼ 10−18 GeV−1, which are well belowour benchmark regions.Despite the fact that neutrino mass spectrum is not completely predictedby the models, the matrix scales in the ISS mechanism are estimated tobe in agreement with the neutrino constraints.1 Moreover, we have nu-merically checked, in all models, that there are solutions with couplingconstants of order one that also satisfy LFV processes and the unitary con-dition. These processes can be easily avoided without fine-tuning in themodels discussed.In subsequent works can be interesting study more in details other LFV pro-cesses, induced in the different ISS models, besides the B(µ→ eγ) consid-ered in the results presented in chap 3, e.g., B(τ→ eγ) or B(τ→ µγ). Ad-ditionally, a phenomenological study of the new exotic fermions invokedto generate the chiral anomaly in the U(1)X symmetry, can be importantto consider, in order to complement the model. Another important idea

1 It is worth mentioning that for all the models the normal spectrum is the preferred neutrinomass spectrum [47] which in turn implies that our results are also compatible with thecosmological upper bound on sum of neutrino masses [269, 270].

C O N C L U S I O N S 69

to consider in the future is studying the implementation of other opera-tors besides the aF eF that imply a photon-axion/ALPs interaction, e.g., acoupling axion/ALPs-fermions or nucleons in order to use other possibleexperimental techniques to search these particles.

In a second case, the vector portal is invoked in order to explain the Beryl-lium anomaly through the decay of a new gauge boson of 17 MeV of massinto electron-positron pairs. Specifically, a new way to search this newgauge boson, so-called dark photons in positron/electron beam-dump ex-periments have been proposed, fixing an important region in the DP pa-rameter space. This region can be tested, in a resonant production mode,by the PADME experiment. We have suggested a new way to search fornarrow resonances, and specifically DP coupled to positron-electron pairs,via resonant production in positron-electron annihilation. This productionmode can be implemented in the Frascati BTF, which is a facility that canprovide positron beams with energy between 250− 550MeV [59]. Thisrange covers precisely the c.m. energy needed to produce via resonantpositron-electron annihilation the mA′ ∼ 17MeV DP invoked to explainthe anomaly observed in 8Be nuclear transitions. By exploiting this pro-duction process, the Frascati PADME experiment will be able to reach wellinside the interesting parameter space region.To close this chapter, we conclude that resonant e+e− → A′ productioncan be relevant also for electron beam dump experiments, since secondarypositrons that could trigger the annihilation process are abundantly pro-duced in electromagnetic showers. This feature has been exploited in re-analysing the SLAC E137 data [60], with the result of extending the pre-viously excluded region towards smaller ε values, as is shown by the lightgray area in Fig. 13. Also, including the two diagrams in Fig. 8-(b) and Fig.8-(c) in the calculation of the exclusion limits for null results of electronbeam-dump experiments and active beam-dump experiments to produceand detect light DM, it is possible to obtain in some selected kinematics upto an order of magnitude gain in sensitivity. In particular, the best exclu-sion limit set by E137 is pushed down by a factor of ∼10 for mDM in therange (20–40 MeV) [61]. These results show that positron annihilationneeds to be included for a correct evaluation of light DM exclusion limitsobtained from electron beam-dump experiments.

AC O N D I T I O N S T O C A N C E L T H E G AU G E A N DG R AV I TAT I O N A L A N O M A L I E S I N T H E Z N S Y M M E T R I E S

Let’s assume that the fields in the model proposed in section 3.1 transformunder a Z N discrete symmetry as:

ψ k → e i 2π Z k / Nψ k , (A.1)

where Z k , with k = qL , dR , uR , L , lR , NR , SR , EL , ER, are integernumbers representing the charges of the fields under the Z N symmetry.The equation to cancel the [SU( 2 )L ]

2 × Z N , [SU( 3 )C ]2 × Z N gauge

anomalies can be written as follows [105–112]:

1. For the [SU( 2 )L ]2 × Z N anomaly, is necessary to sum the charges

under the Z N symmetry taking into account only the SU( 2 )L dou-blet fields. The anomaly equation takes the following form:

A 2 =12

3

3 Z qL+ Z L

= 0 Mod N / 2 , (A.2)

where the 3 outside the parentheses is associated with the familymultiplicity of both quarks and leptons, while the 3 inside the paren-thesis represent a sum over color. The 1

2 factor is the Casimir coeffi-cient associated with SU( 2 )L doublets.

2. For the [SU( 3 )C ]2 × Z N anomaly, the equation can be calculated

taking a sum over the color fields:

A 3 =12

3 ( 2 Z qL− ZuR

− Z dR) = 0 Mod N / 2 , (A.3)

where again 12 corresponds to the Casimir coefficient for fields in the

fundamental representation. The factor 3 is the family multiplicityindex while the factor 2 is associated with the qL SU( 2 )L multiplic-ity. The charges uR and dR have a coefficient equal to one, since theyare SU( 2 )L singlets.

To cancel the gravitational anomaly we consider all the fields present inthe model.

Agrav. = 3 × 3 × 2 Z qL− 3 × 3 × 1 ZuR

− 3 × 3 × 1 Z dR+ 3 × 2 × 1 Z L

−3 × 1 × 1 Z lR− n NR

× 1 × 1 Z NR− n sR

× 1 × 1 ZSR

+ 1 × 1 × 1 Z EL− 1 × 1 × 1 Z ER

= 0 ,

= 1 8 Z qL− 9 ZuR

− 9 Z dR+ 6 Z L − 3 Z lR

− n NRZ NR

−n sRZSR

+ Z EL− Z ER

= 0 Mod N / 2 , (A.4)

71

72 C O N D I T I O N S T O C A N C E L T H E G AU G E A N D G R AV I TAT I O N A L A N O M A L I E S I N T H E zn S Y M M E T R I E S

where the coefficients in front the charges are associated with the family,color, and SU( 2 )L multiplicities. Note that n NR

and nSRrepresent the

number of new singlet leptons added in the ISS mechanism.In this way, replacing the charges for the fields given in table 1 in Eqs.(A.2–A.4), the [SU( 2 )L ]

2 × Z N , [SU( 3 )C ]2 × Z N and [grav. ] 2 × Z N

anomaly coefficients, respectively, are:

(AB2 , AB

3 , ABgrav. ) =

32

, 0 , 0

, (A.5)

(AL2 , AL

3 , ALgrav. ) =

32

, 0 , 3 − n NR− n sR

a + b − c

, (A.6)

where (A2, A3, Agrav.) are respectively the SU(2)L, SU(3)c and gravitationalanomaly coefficients, and B and L refer respectively to the B and L sym-metries, see table 1 in section 3.Writing the ZN discrete symmetry as:

Z = c1B+ c2L, (A.7)

where ci , i = 1, 2, can be rational number with the aim of get ZN chargesintegers, the anomaly equations can be written as:

Ai(Z) = c1ABi + c2AL

i , (A.8)

with i = 2, 3, grav., and the ABi and AL

i coefficients are given in Eqs.(A.5–A.6). Consequently, the [SU(2)L]

2× ZN and [SU(3)C]2× ZN anomaly equa-

tions can be written as:

A2(Z) =32(c1 + c2), (A.9)

A3(Z) = 0. (A.10)

Note that the [SU(3)C]2× ZN anomaly is automatically canceled.

Since the gravitational anomaly coefficients have dependence only on theL symmetry, from Eq.(A.5) and Eq.(A.6), is possible to write the gravita-tional anomaly as:

Agrav(Z) = c2

h

3− nNR− nSR

qd2

+ sdi

, (A.11)

where the definitions a = qd/2 and b = sd + c have been used.

Be+ e− → A′ → e+ e− C R O S S S E C T I O N

According to ref. [261], the cross section for the process e+e− → A′ →e+e− corresponding to diagram (a) in Fig. 16, including 1-loop electro-magnetic corrections and soft photon bremsstrahlung (taking into accountthe finite width of the A′ boson) can be written as:

dσdΩ

= ε4 α2

4s|χ(s)|2 × Cθ × CIR (1+ cΦ+ cS) , (B.1)

where,

χ(s) =s

s−m2A′ + imA′ΓA′

, (B.2)

is the reduced propagator and ΓA′ is the A′ width given by:

ΓA′ = ε2 αmA′

3

√

√

√

1−4m2

e

m2A′

1+2m2

e

m2A′

. (B.3)

The expressions for Cθ and CIR (1+ cΦ+ cS) correspond to bremsstrahlungand virtual electromagnetic corrections, respectively. These are given by: The definition

cθ ≡ Cos(θ ) hasbeen used.

Cθ =

1+8m2

e

s+

1−4m2

e

s

2

c2θ

(B.4)

and

CIR (1+ cΦ+ cS) =

∆EE

1

1− ∆EE χ(s)

2απ βe

× (B.5)

1+2απβe

s−m2A′

mA′ΓA′(Φ−ΦA′)+

α

π

32βe +

13π2−

12

,

where

Φ = arctan

m2A′ − s(1− ∆E

E )

mA′ΓA′

, (B.6)

βe = log s

m2e

−1, ΦA′ = arctan

m2A′ − s

mA′ΓA′

,

with E the energy of the incoming particle and∆E the energy up to whichsoft photons emission is included (∆E/E ∼ 1%).1 In the bremsstrahlung

1 To be more consistent, this should be∆E/Ee, where Ee is the positron energy in the specificcollision, but the difference is irrelevant.

73

74 e+ e− → a ′ → e+ e− C R O S S S E C T I O N

A′ A′

A′A′

γ

γ

γ

e+ e+

e+e+

e+

e+

e−

e− e− e−

e−e−e−

e+

e+

e−

(a) (b)

(c) (d)

FIGURE 16: Feynman diagrams to the process e+e−→ A′→ e+e−: (a) correspondto the tree level process, while (b)-(c) and (d) represent bremsstrahlung andvirtual electromagnetic corrections, respectively.

case (diagram (b) and (c) in Fig. 16) the soft photon can also be radi-ated from the electron leg. Taking the e+ energy to be Ee after t radiationlengths and the electrons inside the target at rest, that is, considering thecenter of mass energy as s = 2me(Ee +me), we obtain the cross section inthe laboratory frame. Nevertheless, as was discussed in section 4.3.1, inorder to take into account the target electron motion it is necessary replacethis Mandelstam variable with the expression given in Eq. (4.9).

B I B L I O G R A P H Y

[1] J. H. Oort, “The force exerted by the stellar system in the direction perpendicular to the galacticplane and some related problems”, Bulletin of the Astronomical Institutes of the Netherlands 6(1932) 249.

[2] M. Roos, “Dark Matter: The evidence from astronomy, astrophysics and cosmology”,arXiv:1001.0316 [astro-ph.CO].

[3] V. C. Rubin, “Dark matter in spiral galaxies”, Scientific American 248 (June, 1983) 96–106.[4] R. Davis, Jr., D. S. Harmer, and K. C. Hoffman, “Search for neutrinos from the sun”, Phys. Rev.

Lett. 20 (1968) 1205–1209.[5] G. Battistoni et al., “Nucleon Stability, Magnetic Monopoles and Atmospheric Neutrinos in the

Mont Blanc Experiment”, Phys. Lett. B133 (1983) 454–460.[6] Daya Bay , F. P. An et al., “Observation of electron-antineutrino disappearance at Daya Bay”,

Phys. Rev. Lett. 108 (2012) 171803, arXiv:1203.1669 [hep-ex].[7] Double Chooz , Y. Abe et al., “Indication for the disappearance of reactor electron antineutrinos

in the Double Chooz experiment”, Phys. Rev. Lett. 108 (2012) 131801, arXiv:1112.6353[hep-ex].

[8] Super-Kamiokande , Y. Fukuda et al., “Evidence for oscillation of atmospheric neutrinos”,Phys.Rev.Lett. 81 (1998) 1562–1567, arXiv:hep-ex/9807003 [hep-ex].

[9] SNO , Q. Ahmad et al., “Direct evidence for neutrino flavor transformation from neutral currentinteractions in the Sudbury Neutrino Observatory”, Phys.Rev.Lett. 89 (2002) 011301,arXiv:nucl-ex/0204008 [nucl-ex].

[10] Y. Hochberg, E. Kuflik, T. Volansky, and J. G. Wacker, “Mechanism for Thermal Relic Dark Matterof Strongly Interacting Massive Particles”, Phys. Rev. Lett. 113 (2014) 171301,arXiv:1402.5143 [hep-ph].

[11] D. E. Kaplan, M. A. Luty, and K. M. Zurek, “Asymmetric Dark Matter”, Phys. Rev. D79 (2009)115016, arXiv:0901.4117 [hep-ph].

[12] G. Jungman, M. Kamionkowski, and K. Griest, “Supersymmetric dark matter”, Phys. Rept. 267(1996) 195–373, arXiv:hep-ph/9506380 [hep-ph].

[13] A. Kusenko, “Sterile neutrinos: The Dark side of the light fermions”, Phys. Rept. 481 (2009)1–28, arXiv:0906.2968 [hep-ph].

[14] B. Carr, M. Raidal, T. Tenkanen, V. Vaskonen, and H. VeermÃ€e, “Primordial black holeconstraints for extended mass functions”, Phys. Rev. D96 no. 2, (2017) 023514,arXiv:1705.05567 [astro-ph.CO].

[15] LUX , D. S. Akerib et al., “Results from a search for dark matter in the complete LUX exposure”,Phys. Rev. Lett. 118 no. 2, (2017) 021303, arXiv:1608.07648 [astro-ph.CO].

[16] PandaX-II , X. Cui et al., “Dark Matter Results From 54-Ton-Day Exposure of PandaX-IIExperiment”, Phys. Rev. Lett. 119 no. 18, (2017) 181302, arXiv:1708.06917 [astro-ph.CO].

[17] XENON , E. Aprile et al., “First Dark Matter Search Results from the XENON1T Experiment”,Phys. Rev. Lett. 119 no. 18, (2017) 181301, arXiv:1705.06655 [astro-ph.CO].

[18] SuperCDMS , R. Agnese et al., “Results from the Super Cryogenic Dark Matter SearchExperiment at Soudan”, Phys. Rev. Lett. 120 no. 6, (2018) 061802, arXiv:1708.08869

75

http://arxiv.org/abs/1001.0316

http://dx.doi.org/10.1038/scientificamerican0683-96

http://dx.doi.org/10.1103/PhysRevLett.20.1205


http://dx.doi.org/10.1016/0370-2693(83)90827-4







http://arxiv.org/abs/hep-ex/9807003


http://arxiv.org/abs/nucl-ex/0204008



http://dx.doi.org/10.1103/PhysRevD.79.115016



http://dx.doi.org/10.1016/0370-1573(95)00058-5

http://dx.doi.org/10.1016/0370-1573(95)00058-5

http://arxiv.org/abs/hep-ph/9506380

http://dx.doi.org/10.1016/j.physrep.2009.07.004














76 B I B L I O G R A P H Y

[hep-ex].[19] DarkSide , P. Agnes et al., “Results From the First Use of Low Radioactivity Argon in a Dark

Matter Search”, Phys. Rev. D93 no. 8, (2016) 081101, arXiv:1510.00702 [astro-ph.CO].[Addendum: Phys. Rev.D95,no.6,069901(2017)].

[20] DEAP-3600 , P. A. Amaudruz et al., “First results from the DEAP-3600 dark matter search withargon at SNOLAB”, Phys. Rev. Lett. 121 no. 7, (2018) 071801, arXiv:1707.08042[astro-ph.CO].

[21] Super-Kamiokande , K. Choi et al., “Search for neutrinos from annihilation of capturedlow-mass dark matter particles in the Sun by Super-Kamiokande”, Phys. Rev. Lett. 114 no. 14,(2015) 141301, arXiv:1503.04858 [hep-ex].

[22] IceCube , M. G. Aartsen et al., “Search for annihilating dark matter in the Sun with 3 years ofIceCube data”, Eur. Phys. J. C77 no. 3, (2017) 146, arXiv:1612.05949 [astro-ph.HE].

[23] Fermi-LAT , M. Ackermann et al., “The Fermi Galactic Center GeV Excess and Implications forDark Matter”, Astrophys. J. 840 no. 1, (2017) 43, arXiv:1704.03910 [astro-ph.HE].

[24] T. Daylan, D. P. Finkbeiner, D. Hooper, T. Linden, S. K. N. Portillo, N. L. Rodd, and T. R. Slatyer,“The characterization of the gamma-ray signal from the central Milky Way: A case forannihilating dark matter”, Phys. Dark Univ. 12 (2016) 1–23, arXiv:1402.6703 [astro-ph.HE].

[25] Fermi-LAT , M. Ackermann et al., “Searching for Dark Matter Annihilation from Milky WayDwarf Spheroidal Galaxies with Six Years of Fermi Large Area Telescope Data”, Phys. Rev. Lett.115 no. 23, (2015) 231301, arXiv:1503.02641 [astro-ph.HE].

[26] G. Steigman, “CMB Constraints On The Thermal WIMP Mass And Annihilation Cross Section”,Phys. Rev. D91 no. 8, (2015) 083538, arXiv:1502.01884 [astro-ph.CO].

[27] A. Cuoco, M. KrÃ€mer, and M. Korsmeier, “Novel Dark Matter Constraints from Antiprotons inLight of AMS-02”, Phys. Rev. Lett. 118 no. 19, (2017) 191102, arXiv:1610.03071[astro-ph.HE].

[28] M.-Y. Cui, Q. Yuan, Y.-L. S. Tsai, and Y.-Z. Fan, “Possible dark matter annihilation signal in theAMS-02 antiproton data”, Phys. Rev. Lett. 118 no. 19, (2017) 191101, arXiv:1610.03840[astro-ph.HE].

[29] L. Bergstrom, T. Bringmann, I. Cholis, D. Hooper, and C. Weniger, “New limits on dark matterannihilation from AMS cosmic ray positron data”, Phys. Rev. Lett. 111 (2013) 171101,arXiv:1306.3983 [astro-ph.HE].

[30] A. Birkedal, K. Matchev, and M. Perelstein, “Dark matter at colliders: A Model independentapproach”, Phys. Rev. D70 (2004) 077701, arXiv:hep-ph/0403004 [hep-ph].

[31] M. Beltran, D. Hooper, E. W. Kolb, Z. A. C. Krusberg, and T. M. P. Tait, “Maverick dark matter atcolliders”, JHEP 09 (2010) 037, arXiv:1002.4137 [hep-ph].

[32] A. Boveia and C. Doglioni, “Dark Matter Searches at Colliders”, Ann. Rev. Nucl. Part. Sci. 68(2018) 429–459, arXiv:1810.12238 [hep-ex].

[33] B. Penning, “The pursuit of dark matter at collidersâan overview”, J. Phys. G45 no. 6, (2018)063001, arXiv:1712.01391 [hep-ex].

[34] J. Alexander et al., “Dark Sectors 2016 Workshop: Community Report”, arXiv:1608.08632[hep-ph]. http://inspirehep.net/record/1484628/files/arXiv:1608.08632.pdf.

[35] R. Essig et al., “Working Group Report: New Light Weakly Coupled Particles”, in Proceedings,2013 Community Summer Study on the Future of U.S. Particle Physics: Snowmass on theMississippi (CSS2013): Minneapolis, MN, USA, July 29-August 6, 2013. 2013. arXiv:1311.0029[hep-ph]. http://inspirehep.net/record/1263039/files/arXiv:1311.0029.pdf.



http://dx.doi.org/10.1103/PhysRevD.93.081101, 10.1103/PhysRevD.95.069901








http://dx.doi.org/10.1140/epjc/s10052-017-4689-9


http://dx.doi.org/10.3847/1538-4357/aa6cab


http://dx.doi.org/10.1016/j.dark.2015.12.005

















http://dx.doi.org/10.1007/JHEP09(2010)037


http://dx.doi.org/10.1146/annurev-nucl-101917-021008



http://dx.doi.org/10.1088/1361-6471/aabea7

http://dx.doi.org/10.1088/1361-6471/aabea7




http://inspirehep.net/record/1484628/files/arXiv:1608.08632.pdf



http://inspirehep.net/record/1263039/files/arXiv:1311.0029.pdf


[36] Fundamental Physics at the Intensity Frontier. 2012. arXiv:1205.2671 [hep-ex].https://inspirehep.net/record/1114323/files/arXiv:1205.2671.pdf.

[37] M. Battaglieri et al., “US Cosmic Visions: New Ideas in Dark Matter 2017: Community Report”,arXiv:1707.04591 [hep-ph].

[38] J. Jaeckel and A. Ringwald, “The Low-Energy Frontier of Particle Physics”, Ann.Rev.Nucl.Part.Sci.60 (2010) 405–437, arXiv:1002.0329 [hep-ph].

[39] R. D. Peccei and H. R. Quinn, “CP Conservation in the Presence of Instantons”, Phys. Rev. Lett.38 (1977) 1440–1443.

[40] R. D. Peccei and H. R. Quinn, “Constraints imposed by CP conservation in the presence ofpseudoparticles”, Phys. Rev. D 16 (Sep, 1977) 1791–1797.https://link.aps.org/doi/10.1103/PhysRevD.16.1791.

[41] S. Weinberg, “A New Light Boson?”, Phys. Rev. Lett. 40 (1978) 223–226.[42] F. Wilczek, “Problem of Strong P and T Invariance in the Presence of Instantons”, Phys. Rev. Lett.

40 (1978) 279–282.[43] A. Ringwald, “Exploring the Role of Axions and Other WISPs in the Dark Universe”, Phys. Dark

Univ. 1 (2012) 116–135, arXiv:1210.5081 [hep-ph].[44] A. G. Dias, A. C. B. Machado, C. C. Nishi, A. Ringwald, and P. Vaudrevange, “The Quest for an

Intermediate-Scale Accidental Axion and Further ALPs”, JHEP 1406 (2014) 037,arXiv:1403.5760 [hep-ph].

[45] R. N. Mohapatra, “Mechanism for Understanding Small Neutrino Mass in Superstring Theories”,Phys. Rev. Lett. 56 (1986) 561–563.

[46] R. N. Mohapatra and J. W. F. Valle, “Neutrino Mass and Baryon Number Nonconservation inSuperstring Models”, Phys. Rev. D34 (1986) 1642.

[47] A. Abada and M. Lucente, “Looking for the minimal inverse seesaw realisation”, Nucl. Phys.B885 (2014) 651–678, arXiv:1401.1507 [hep-ph].

[48] A. Abada, G. Arcadi, and M. Lucente, “Dark Matter in the minimal Inverse Seesaw mechanism”,JCAP 1410 (2014) 001, arXiv:1406.6556 [hep-ph].

[49] C. D. R. Carvajal, B. L. Sánchez-Vega, and O. Zapata, “Linking axionlike dark matter to neutrinomasses”, Phys. Rev. D96 no. 11, (2017) 115035, arXiv:1704.08340 [hep-ph].

[50] T. Blum, A. Denig, I. Logashenko, E. de Rafael, B. Lee Roberts, T. Teubner, and G. Venanzoni,“The Muon (g-2) Theory Value: Present and Future”, arXiv:1311.2198 [hep-ph].

[51] A. J. Krasznahorkay et al., “Observation of Anomalous Internal Pair Creation in Be8 : A PossibleIndication of a Light, Neutral Boson”, Phys. Rev. Lett. 116 no. 4, (2016) 042501,arXiv:1504.01527 [nucl-ex].

[52] A. J. Krasznahorkay et al., “On the creation of the 17 MeV X boson in the 17.6 MeV M1transition of 8Be”, EPJ Web Conf. 142 (2017) 01019.

[53] A. J. Krasznahorkay et al., “New experimental results for the 17 MeV particle created in 8Be”,EPJ Web Conf. 137 (2017) 08010.

[54] M. Pospelov, “Secluded U(1) below the weak scale”, Phys. Rev. D80 (2009) 095002,arXiv:0811.1030 [hep-ph].

[55] J. L. Feng, B. Fornal, I. Galon, S. Gardner, J. Smolinsky, T. M. P. Tait, and P. Tanedo, “ProtophobicFifth-Force Interpretation of the Observed Anomaly in 8Be Nuclear Transitions”, Phys. Rev. Lett.117 no. 7, (2016) 071803, arXiv:1604.07411 [hep-ph].

[56] B. Holdom, “Two U(1)’s and Epsilon Charge Shifts”, Phys. Lett. 166B (1986) 196–198.[57] J. Jaeckel, “A force beyond the Standard Model - Status of the quest for hidden photons”,

Frascati Phys. Ser. 56 (2012) 172–192, arXiv:1303.1821 [hep-ph].

http://dx.doi.org/10.2172/1042577


https://inspirehep.net/record/1114323/files/arXiv:1205.2671.pdf


http://dx.doi.org/10.1146/annurev.nucl.012809.104433

http://dx.doi.org/10.1146/annurev.nucl.012809.104433





https://link.aps.org/doi/10.1103/PhysRevD.16.1791











http://dx.doi.org/10.1016/j.nuclphysb.2014.06.003



http://dx.doi.org/10.1088/1475-7516/2014/10/001







http://dx.doi.org/10.1051/epjconf/201714201019







http://dx.doi.org/10.1016/0370-2693(86)91377-8



[58] M. Raggi and V. Kozhuharov, “Results and perspectives in dark photon physics”, Riv. Nuovo Cim.38 no. 10, (2015) 449–505.

[59] E. Nardi, C. D. R. Carvajal, A. Ghoshal, D. Meloni, and M. Raggi, “Resonant production of darkphotons in positron beam dump experiments”, Phys. Rev. D 97 (May, 2018) 095004.https://link.aps.org/doi/10.1103/PhysRevD.97.095004.

[60] L. Marsicano, M. Battaglieri, M. Bondí, C. D. R. Carvajal, A. Celentano, M. De Napoli, R. De Vita,E. Nardi, M. Raggi, and P. Valente, “Dark photon production through positron annihilation inbeam-dump experiments”, Phys. Rev. D 98 (Jul, 2018) 015031.https://link.aps.org/doi/10.1103/PhysRevD.98.015031.

[61] L. Marsicano, M. Battaglieri, M. Bondí, C. D. R. Carvajal, A. Celentano, M. De Napoli, R. De Vita,E. Nardi, M. Raggi, and P. Valente, “Novel Way to Search for Light Dark Matter in LeptonBeam-Dump Experiments”, Phys. Rev. Lett. 121 (Jul, 2018) 041802.https://link.aps.org/doi/10.1103/PhysRevLett.121.041802.

[62] B. A. Dobrescu, “Massless gauge bosons other than the photon”, Phys. Rev. Lett. 94 (2005)151802, arXiv:hep-ph/0411004 [hep-ph].

[63] R. D. Peccei, “The Strong CP problem and axions”, Lect. Notes Phys. 741 (2008) 3–17,arXiv:hep-ph/0607268 [hep-ph].

[64] W. A. Bardeen, “Anomalous Currents in Gauge Field Theories”, Nucl. Phys. B75 (1974) 246–258.[65] G. ’t Hooft, “Symmetry Breaking Through Bell-Jackiw Anomalies”, Phys. Rev. Lett. 37 (1976)

8–11.[66] G. ’t Hooft, “Computation of the Quantum Effects Due to a Four-Dimensional Pseudoparticle”,

Phys. Rev. D14 (1976) 3432–3450. [Erratum: Phys. Rev.D18,2199(1978)].[67] C. Callan, R. Dashen, and D. Gross, “The structure of the gauge theory vacuum”, Physics Letters

B 63 no. 3, (1976) 334 – 340.http://www.sciencedirect.com/science/article/pii/037026937690277X.

[68] R. Jackiw and C. Rebbi, “Vacuum Periodicity in a Yang-Mills Quantum Theory”, Phys. Rev. Lett.37 (Jul, 1976) 172–175. https://link.aps.org/doi/10.1103/PhysRevLett.37.172.

[69] V. Baluni, “CP-nonconserving effects in quantum chromodynamics”, Phys. Rev. D 19 (Apr, 1979)2227–2230. https://link.aps.org/doi/10.1103/PhysRevD.19.2227.

[70] R. Crewther, P. D. Vecchia, G. Veneziano, and E. Witten, “Chiral estimate of the electric dipolemoment of the neutron in quantum chromodynamics”, Physics Letters B 88 no. 1, (1979) 123 –127. http://www.sciencedirect.com/science/article/pii/037026937990128X.

[71] J. M. Pendlebury et al., “Revised experimental upper limit on the electric dipole moment of theneutron”, Phys. Rev. D92 no. 9, (2015) 092003, arXiv:1509.04411 [hep-ex].

[72] C. A. Baker et al., “An Improved experimental limit on the electric dipole moment of theneutron”, Phys. Rev. Lett. 97 (2006) 131801, arXiv:hep-ex/0602020 [hep-ex].

[73] S. Aoki et al., “Review of lattice results concerning low-energy particle physics”, Eur. Phys. J. C77no. 2, (2017) 112, arXiv:1607.00299 [hep-lat].

[74] ParticleDataGroup , M. Tanabashi et al., “Review of Particle Physics”, Phys. Rev. D98 no. 3,(2018) 030001.

[75] A. Nelson, “Naturally weak CP violation”, Physics Letters B 136 no. 5, (1984) 387 – 391.http://www.sciencedirect.com/science/article/pii/0370269384920252.

[76] S. M. Barr, “Solving the Strong CP Problem without the Peccei-Quinn Symmetry”, Phys. Rev. Lett.53 (Jul, 1984) 329–332. https://link.aps.org/doi/10.1103/PhysRevLett.53.329.

[77] M. Dine and P. Draper, “Challenges for the Nelson-Barr Mechanism”, JHEP 08 (2015) 132,arXiv:1506.05433 [hep-ph].

http://dx.doi.org/10.1393/ncr/i2015-10117-9

http://dx.doi.org/10.1393/ncr/i2015-10117-9






https://link.aps.org/doi/10.1103/PhysRevLett.121.041802




http://dx.doi.org/10.1007/978-3-540-73518-2_1


http://dx.doi.org/10.1016/0550-3213(74)90546-X



http://dx.doi.org/10.1103/PhysRevD.18.2199.3, 10.1103/PhysRevD.14.3432

http://dx.doi.org/https://doi.org/10.1016/0370-2693(76)90277-X


http://www.sciencedirect.com/science/article/pii/037026937690277X









http://www.sciencedirect.com/science/article/pii/037026937990128X










http://dx.doi.org/https://doi.org/10.1016/0370-2693(84)92025-2

http://www.sciencedirect.com/science/article/pii/0370269384920252







[78] G. G. Raffelt, “Astrophysical methods to constrain axions and other novel particle phenomena”,Phys. Rept. 198 (1990) 1–113.

[79] M. Kuster, G. Raffelt, and B. Beltran, “Axions: Theory, cosmology, and experimental searches.Proceedings, 1st Joint ILIAS-CERN-CAST axion training, Geneva, Switzerland, November30-December 2, 2005”, Lect. Notes Phys. 741 (2008) pp.1–258.

[80] D. J. E. Marsh, “Axion Cosmology”, Phys. Rept. 643 (2016) 1–79, arXiv:1510.07633[astro-ph.CO].

[81] M. Srednicki, “Axion Couplings to Matter. 1. CP Conserving Parts”, Nucl. Phys. B260 (1985)689–700.

[82] J. E. Kim, “Light pseudoscalars, particle physics and cosmology”, Physics Reports 150 no. 1,(1987) 1 – 177. http://www.sciencedirect.com/science/article/pii/0370157387900172.

[83] W. A. Bardeen, R. D. Peccei, and T. Yanagida, “CONSTRAINTS ON VARIANT AXION MODELS”,Nucl. Phys. B279 (1987) 401–428.

[84] P. Arias, D. Cadamuro, M. Goodsell, J. Jaeckel, J. Redondo, et al., “WISPy Cold Dark Matter”,JCAP 1206 (2012) 013, arXiv:1201.5902 [hep-ph].

[85] P. Sikivie, “Axion Cosmology”, Lect. Notes Phys. 741 (2008) 19–50, arXiv:astro-ph/0610440[astro-ph]. [,19(2006)].

[86] G. Grilli di Cortona, E. Hardy, J. Pardo Vega, and G. Villadoro, “The QCD axion, precisely”, JHEP01 (2016) 034, arXiv:1511.02867 [hep-ph].

[87] O. Wantz and E. P. S. Shellard, “Axion cosmology revisited”, Phys. Rev. D 82 (Dec, 2010) 123508.http://link.aps.org/doi/10.1103/PhysRevD.82.123508.

[88] J. A. Grifols, E. Masso, and R. Toldra, “Gamma-rays from SN1987A due to pseudoscalarconversion”, Phys.Rev.Lett. 77 (1996) 2372–2375, arXiv:astro-ph/9606028 [astro-ph].

[89] J. W. Brockway, E. D. Carlson, and G. G. Raffelt, “SN1987A gamma-ray limits on the conversionof pseudoscalars”, Phys.Lett. B383 (1996) 439–443, arXiv:astro-ph/9605197 [astro-ph].

[90] M. Kuster, G. Raffelt, and B. Beltran, Astrophysical Axion Bounds, pp. 51–71. Springer BerlinHeidelberg, Berlin, Heidelberg, 2008. https://doi.org/10.1007/978-3-540-73518-2_3.

[91] A. Arvanitaki, S. Dimopoulos, S. Dubovsky, N. Kaloper, and J. March-Russell, “String Axiverse”,Phys. Rev. D81 (2010) 123530, arXiv:0905.4720 [hep-th].

[92] A. Arvanitaki and S. Dubovsky, “Exploring the String Axiverse with Precision Black HolePhysics”, Phys. Rev. D83 (2011) 044026, arXiv:1004.3558 [hep-th].

[93] A. Arvanitaki, M. Baryakhtar, and X. Huang, “Discovering the QCD Axion with Black Holes andGravitational Waves”, Phys. Rev. D91 no. 8, (2015) 084011, arXiv:1411.2263 [hep-ph].

[94] R. D. Peccei, “The Strong CP Problem”, Adv. Ser. Direct. High Energy Phys. 3 (1989) 503–551.[95] M. S. Turner, “Windows on the Axion”, Phys. Rept. 197 (1990) 67–97.[96] D. S. M. Alves and N. Weiner, “A viable QCD axion in the MeV mass range”, JHEP 07 (2018)

092, arXiv:1710.03764 [hep-ph].[97] E. Izaguirre, T. Lin, and B. Shuve, “Searching for Axionlike Particles in Flavor-Changing Neutral

Current Processes”, Phys. Rev. Lett. 118 no. 11, (2017) 111802, arXiv:1611.09355 [hep-ph].[98] J. E. Kim, “Weak Interaction Singlet and Strong CP Invariance”, Phys. Rev. Lett. 43 (1979) 103.[99] M. A. Shifman, A. I. Vainshtein, and V. I. Zakharov, “Can Confinement Ensure Natural CP

Invariance of Strong Interactions?”, Nucl. Phys. B166 (1980) 493–506.[100] L. Di Luzio, F. Mescia, and E. Nardi, “Redefining the Axion Window”, Phys. Rev. Lett. 118 (Jan,

2017) 031801. https://link.aps.org/doi/10.1103/PhysRevLett.118.031801.[101] S. L. Cheng, C. Q. Geng, and W. T. Ni, “Axion - photon couplings in invisible axion models”, Phys.

Rev. D52 (1995) 3132–3135, arXiv:hep-ph/9506295 [hep-ph].

http://dx.doi.org/10.1016/0370-1573(90)90054-6




http://dx.doi.org/10.1016/0550-3213(85)90054-9

http://dx.doi.org/10.1016/0550-3213(85)90054-9




http://dx.doi.org/10.1016/0550-3213(87)90003-4

http://dx.doi.org/10.1088/1475-7516/2012/06/013


http://dx.doi.org/10.1007/978-3-540-73518-2_2

http://arxiv.org/abs/astro-ph/0610440






http://link.aps.org/doi/10.1103/PhysRevD.82.123508



http://dx.doi.org/10.1016/0370-2693(96)00778-2


http://dx.doi.org/10.1007/978-3-540-73518-2_3

https://doi.org/10.1007/978-3-540-73518-2_3







http://dx.doi.org/10.1142/9789814503280_0013

http://dx.doi.org/10.1016/0370-1573(90)90172-X







http://dx.doi.org/10.1016/0550-3213(80)90209-6








[102] P. W. Graham and S. Rajendran, “New Observables for Direct Detection of Axion Dark Matter”,Phys. Rev. D88 (2013) 035023, arXiv:1306.6088 [hep-ph].

[103] A. R. Zhitnitsky, “On Possible Suppression of the Axion Hadron Interactions. (In Russian)”, Sov.J. Nucl. Phys. 31 (1980) 260. [Yad. Fiz.31,497(1980)].

[104] M. Dine, W. Fischler, and M. Srednicki, “A Simple Solution to the Strong CP Problem with aHarmless Axion”, Phys. Lett. B104 (1981) 199–202.

[105] L. M. Krauss and F. Wilczek, “Discrete gauge symmetry in continuum theories”, Phys. Rev. Lett.62 (Mar, 1989) 1221–1223. http://link.aps.org/doi/10.1103/PhysRevLett.62.1221.

[106] L. Ibáñez and G. Ross, “Discrete gauge symmetry anomalies”, Physics Letters B 260 no. 3-4,(1991) 291 – 295.http://www.sciencedirect.com/science/article/pii/0370269391916142.

[107] T. Banks and M. Dine, “Note on discrete gauge anomalies”, Phys. Rev. D 45 (Feb, 1992)1424–1427. http://link.aps.org/doi/10.1103/PhysRevD.45.1424.

[108] L. E. Ibáñez, “More about discrete gauge anomalies”, Nucl. Phys. B398 (1993) 301–318,arXiv:hep-ph/9210211 [hep-ph].

[109] C. Luhn and P. Ramond, “Anomaly Conditions for Non-Abelian Finite Family Symmetries”, JHEP07 (2008) 085, arXiv:0805.1736 [hep-ph].

[110] P. Ramond, “Mass hierarchies from anomalies: A Peek behind the Planck curtain”, in Frontiers inquantum field theory. Proceedings, International Conference in Honor of Keiji Kikkawa’s 60thBirthday, Toyonaka, Japan, December 14-17, 1995, pp. 285–300. 1995. arXiv:hep-ph/9604251[hep-ph].

[111] C. Csaki and H. Murayama, “Discrete anomaly matching”, Nucl. Phys. B515 (1998) 114–162,arXiv:hep-th/9710105 [hep-th].

[112] T. Araki, T. Kobayashi, J. Kubo, S. Ramos-Sanchez, M. Ratz, and P. K. S. Vaudrevange,“(Non-)Abelian discrete anomalies”, Nucl. Phys. B805 (2008) 124–147, arXiv:0805.0207[hep-th].

[113] A. Ringwald, “Axions and Axion-Like Particles”, in Proceedings, 49th Rencontres de Moriond onElectroweak Interactions and Unified Theories: La Thuile, Italy, March 15-22, 2014, pp. 223–230.2014. arXiv:1407.0546 [hep-ph].https://inspirehep.net/record/1304441/files/arXiv:1407.0546.pdf.

[114] A. Mirizzi, G. G. Raffelt, and P. D. Serpico, “Signatures of axion-like particles in the spectra ofTeV gamma-ray sources”, Phys.Rev. D76 (2007) 023001, arXiv:0704.3044 [astro-ph].

[115] E. Masso, “Axions and their relatives”, Lect. Notes Phys. 741 (2008) 83–94,arXiv:hep-ph/0607215 [hep-ph].

[116] P. Sikivie, “Experimental Tests of the Invisible Axion”, Phys. Rev. Lett. 51 (1983) 1415–1417.[Erratum: Phys. Rev. Lett.52,695(1984)].

[117] D. A. Dicus, E. W. Kolb, V. L. Teplitz, and R. V. Wagoner, “Astrophysical Bounds on the Masses ofAxions and Higgs Particles”, Phys. Rev. D18 (1978) 1829.

[118] G. Raffelt and L. Stodolsky, “Mixing of the Photon with Low Mass Particles”, Phys. Rev. D37(1988) 1237.

[119] F. Tavecchio, M. Roncadelli, and G. Galanti, “Photons to axion-like particles conversion in ActiveGalactic Nuclei”, arXiv:1406.2303 [astro-ph.HE].

[120] M. Kamionkowski and J. March-Russell, “Planck scale physics and the Peccei-Quinn mechanism”,Phys. Lett. B282 (1992) 137–141, arXiv:hep-th/9202003 [hep-th].

[121] S. M. Barr and D. Seckel, “Planck scale corrections to axion models”, Phys. Rev. D46 (1992)539–549.



http://dx.doi.org/10.1016/0370-2693(81)90590-6



http://link.aps.org/doi/10.1103/PhysRevLett.62.1221

http://dx.doi.org/10.1016/0370-2693(91)91614-2

http://dx.doi.org/10.1016/0370-2693(91)91614-2





http://dx.doi.org/10.1016/0550-3213(93)90111-2


http://dx.doi.org/10.1088/1126-6708/2008/07/085

http://dx.doi.org/10.1088/1126-6708/2008/07/085




http://dx.doi.org/10.1016/S0550-3213(97)00839-0

http://arxiv.org/abs/hep-th/9710105





https://inspirehep.net/record/1304441/files/arXiv:1407.0546.pdf



http://dx.doi.org/10.1007/978-3-540-73518-2_5


http://dx.doi.org/10.1103/PhysRevLett.51.1415, 10.1103/PhysRevLett.52.695.2





http://dx.doi.org/10.1016/0370-2693(92)90492-M

http://arxiv.org/abs/hep-th/9202003




[122] S. Hannestad, A. Mirizzi, G. G. Raffelt, and Y. Y. Y. Wong, “Neutrino and axion hot dark matterbounds after WMAP-7”, JCAP 1008 (2010) 001, arXiv:1004.0695 [astro-ph.CO].

[123] A. E. Nelson and J. Scholtz, “Dark Light, Dark Matter and the Misalignment Mechanism”, Phys.Rev. D84 (2011) 103501, arXiv:1105.2812 [hep-ph].

[124] P. Sikivie and Q. Yang, “Bose-Einstein Condensation of Dark Matter Axions”, Phys. Rev. Lett. 103(Sep, 2009) 111301. http://link.aps.org/doi/10.1103/PhysRevLett.103.111301.

[125] D. Cadamuro and J. Redondo, “Cosmological bounds on pseudo Nambu-Goldstone bosons”,JCAP 1202 (2012) 032, arXiv:1110.2895 [hep-ph].

[126] M. Millea, L. Knox, and B. Fields, “New Bounds for Axions and Axion-Like Particles withkeV-GeV Masses”, Phys. Rev. D92 no. 2, (2015) 023010, arXiv:1501.04097 [astro-ph.CO].

[127] M. Bauer, M. Heiles, M. Neubert, and A. Thamm, “Axion-Like Particles at Future Colliders”,arXiv:1808.10323 [hep-ph].

[128] M. Bauer, M. Neubert, and A. Thamm, “Collider Probes of Axion-Like Particles”,arXiv:1708.00443 [hep-ph].

[129] M. Bauer, M. Neubert, and A. Thamm, “LHC as an Axion Factory: Probing an Axion Explanationfor ( g − 2)µ with Exotic Higgs Decays”, Phys. Rev. Lett. 119 no. 3, (2017) 031802,arXiv:1704.08207 [hep-ph].

[130] B. A. Dobrescu, G. L. Landsberg, and K. T. Matchev, “Higgs boson decays to CP odd scalars at theTevatron and beyond”, Phys. Rev. D63 (2001) 075003, arXiv:hep-ph/0005308 [hep-ph].

[131] D. Curtin et al., “Exotic decays of the 125 GeV Higgs boson”, Phys. Rev. D90 no. 7, (2014)075004, arXiv:1312.4992 [hep-ph].

[132] K. Mimasu and V. Sanz, “ALPs at Colliders”, JHEP 06 (2015) 173, arXiv:1409.4792 [hep-ph].[133] J. Jaeckel and M. Spannowsky, “Probing MeV to 90 GeV axion-like particles with LEP and LHC”,

Phys. Lett. B753 (2016) 482–487, arXiv:1509.00476 [hep-ph].[134] S. Knapen, T. Lin, H. K. Lou, and T. Melia, “Searching for Axionlike Particles with

Ultraperipheral Heavy-Ion Collisions”, Phys. Rev. Lett. 118 no. 17, (2017) 171801,arXiv:1607.06083 [hep-ph].

[135] I. Brivio, M. B. Gavela, L. Merlo, K. Mimasu, J. M. No, R. del Rey, and V. Sanz, “ALPs EffectiveField Theory and Collider Signatures”, arXiv:1701.05379 [hep-ph].

[136] E. M. Riordan et al., “A Search for Short Lived Axions in an Electron Beam Dump Experiment”,Phys. Rev. Lett. 59 (1987) 755.

[137] J. D. Bjorken, S. Ecklund, W. R. Nelson, A. Abashian, C. Church, B. Lu, L. W. Mo, T. A.Nunamaker, and P. Rassmann, “Search for Neutral Metastable Penetrating Particles Produced inthe SLAC Beam Dump”, Phys. Rev. D38 (1988) 3375.

[138] S. Alekhin et al., “A facility to Search for Hidden Particles at the CERN SPS: the SHiP physicscase”, Rept. Prog. Phys. 79 no. 12, (2016) 124201, arXiv:1504.04855 [hep-ph].

[139] B. DÃ¶brich, J. Jaeckel, F. Kahlhoefer, A. Ringwald, and K. Schmidt-Hoberg, “ALPtraum: ALPproduction in proton beam dump experiments”, JHEP 02 (2016) 018, arXiv:1512.03069[hep-ph]. [JHEP02,018(2016)].

[140] D. Budker, P. W. Graham, M. Ledbetter, S. Rajendran, and A. Sushkov, “Proposal for a CosmicAxion Spin Precession Experiment (CASPEr)”, Phys. Rev. X4 no. 2, (2014) 021030,arXiv:1306.6089 [hep-ph].

[141] P. W. Graham, I. G. Irastorza, S. K. Lamoreaux, A. Lindner, and K. A. van Bibber, “ExperimentalSearches for the Axion and Axion-Like Particles”, Ann. Rev. Nucl. Part. Sci. 65 (2015) 485–514,arXiv:1602.00039 [hep-ex].

http://dx.doi.org/10.1088/1475-7516/2010/08/001







http://link.aps.org/doi/10.1103/PhysRevLett.103.111301

http://dx.doi.org/10.1088/1475-7516/2012/02/032















http://dx.doi.org/10.1016/j.physletb.2015.12.037







http://dx.doi.org/10.1088/0034-4885/79/12/124201





http://dx.doi.org/10.1103/PhysRevX.4.021030





[142] E. Armengaud et al., “Axion searches with the EDELWEISS-II experiment”, JCAP 1311 (2013)067, arXiv:1307.1488 [astro-ph.CO].

[143] R. Essig, R. Harnik, J. Kaplan, and N. Toro, “Discovering New Light States at NeutrinoExperiments”, Phys. Rev. D82 (2010) 113008, arXiv:1008.0636 [hep-ph].

[144] BaBar , J. P. Lees et al., “Search for a muonic dark force at BABAR”, Phys. Rev. D94 no. 1, (2016)011102, arXiv:1606.03501 [hep-ex].

[145] A. Friedland, M. Giannotti, and M. Wise, “Constraining the Axion-Photon Coupling with MassiveStars”, Phys.Rev.Lett. 110 no. 6, (2013) 061101, arXiv:1210.1271 [hep-ph].

[146] A. Payez, C. Evoli, T. Fischer, M. Giannotti, A. Mirizzi, et al., “Revisiting the SN1987A gamma-raylimit on ultralight axion-like particles”, JCAP 1502 no. 02, (2015) 006, arXiv:1410.3747[astro-ph.HE].

[147] R. Bähre, B. Döbrich, J. Dreyling-Eschweiler, S. Ghazaryan, R. Hodajerdi, et al., “Any lightparticle search II -Technical Design Report”, JINST 8 (2013) T09001, arXiv:1302.5647[physics.ins-det].

[148] CAST , M. Arik, E. Aune, et al., “New solar axion search using the CERN Axion Solar Telescopewith 4He filling”, Phys. Rev. D 92 (Jul, 2015) 021101.http://link.aps.org/doi/10.1103/PhysRevD.92.021101.

[149] E. Armengaud, F. Avignone, M. Betz, P. Brax, P. Brun, et al., “Conceptual Design of theInternational Axion Observatory (IAXO)”, JINST 9 (2014) T05002, arXiv:1401.3233[physics.ins-det].

[150] L. J. Rosenberg, “Dark-matter QCD-axion searches”, J. Phys. Conf. Ser. 203 (2010) 012008.[151] M. Meyer, D. Horns, and M. Raue, “First lower limits on the photon-axion-like particle coupling

from very high energy gamma-ray observations”, Phys.Rev. D87 no. 3, (2013) 035027,arXiv:1302.1208 [astro-ph.HE].

[152] J. P. Conlon and M. D. Marsh, “Excess Astrophysical Photons from a 0.1 − 1 keV Cosmic AxionBackground”, Phys.Rev.Lett. 111 no. 15, (2013) 151301, arXiv:1305.3603 [astro-ph.CO].

[153] S. Angus, J. P. Conlon, M. C. D. Marsh, A. J. Powell, and L. T. Witkowski, “Soft X-ray Excess inthe Coma Cluster from a Cosmic Axion Background”, JCAP 1409 no. 09, (2014) 026,arXiv:1312.3947 [astro-ph.HE].

[154] T. Higaki, K. S. Jeong, and F. Takahashi, “The 7 keV axion dark matter and the X-ray line signal”,Phys.Lett. B733 (2014) 25–31, arXiv:1402.6965 [hep-ph].

[155] J. Jaeckel, J. Redondo, and A. Ringwald, “3.55 keV hint for decaying axionlike particle darkmatter”, Phys.Rev. D89 (2014) 103511, arXiv:1402.7335 [hep-ph].

[156] Y. Kahn, B. R. Safdi, and J. Thaler, “Broadband and Resonant Approaches to Axion Dark MatterDetection”, Phys. Rev. Lett. 117 no. 14, (2016) 141801, arXiv:1602.01086 [hep-ph].

[157] ALPS , K. Ehret, “The ALPS Light Shining Through a Wall Experiment - WISP Search in theLaboratory”, arXiv:1006.5741 [hep-ex].

[158] K. Ehret, M. Frede, S. Ghazaryan, M. Hildebrandt, E.-A. Knabbe, et al., “New ALPS Results onHidden-Sector Lightweights”, Phys.Lett. B689 (2010) 149–155, arXiv:1004.1313 [hep-ex].

[159] J. Redondo and A. Ringwald, “Light shining through walls”, Contemp.Phys. 52 (2011) 211–236,arXiv:1011.3741 [hep-ph].

[160] F. Januschek, “Light-shining-through-walls with lasers”, arXiv:1410.1633[physics.ins-det].

[161] I. P. Stern, “Axion Dark Matter Searches”, AIP Conf.Proc. 1604 (2014) 456–461,arXiv:1403.5332 [physics.ins-det].

http://dx.doi.org/10.1088/1475-7516/2013/11/067

http://dx.doi.org/10.1088/1475-7516/2013/11/067









http://dx.doi.org/10.1088/1475-7516/2015/02/006



http://dx.doi.org/10.1088/1748-0221/8/09/T09001





http://dx.doi.org/10.1088/1748-0221/9/05/T05002



http://dx.doi.org/10.1088/1742-6596/203/1/012008





http://dx.doi.org/10.1088/1475-7516/2014/09/026











http://dx.doi.org/10.1080/00107514.2011.563516




http://dx.doi.org/10.1063/1.4883465



[162] T. Shokair, J. Root, K. Van Bibber, B. Brubaker, Y. Gurevich, et al., “Future Directions in theMicrowave Cavity Search for Dark Matter Axions”, Int.J.Mod.Phys. A29 (2014) 1443004,arXiv:1405.3685 [physics.ins-det].

[163] CAST , S. Aune et al., “CAST search for sub-eV mass solar axions with 3He buffer gas”,Phys.Rev.Lett. 107 (2011) 261302, arXiv:1106.3919 [hep-ex].

[164] CAST , S. Andriamonje et al., “An Improved limit on the axion-photon coupling from the CASTexperiment”, JCAP 0704 (2007) 010, arXiv:hep-ex/0702006 [hep-ex].

[165] IAXO, CAST , T. Dafni and F. J. Iguaz, “Axion helioscopes update: the status of CAST & IAXO”,PoS TIPP2014 (2014) 130, arXiv:1501.01456 [physics.ins-det].

[166] I. Irastorza, F. Avignone, S. Caspi, J. Carmona, T. Dafni, et al., “Towards a new generation axionhelioscope”, JCAP 1106 (2011) 013, arXiv:1103.5334 [hep-ex].

[167] R. Essig, J. Kaplan, P. Schuster, and N. Toro, “On the Origin of Light Dark Matter Species”,Submitted to: Physical Review D (2010) 1237, arXiv:1004.0691 [hep-ph].

[168] F. del Aguila, G. D. Coughlan, and M. Quiros, “Gauge Coupling Renormalization With SeveralU(1) Factors”, Nucl. Phys. B307 (1988) 633. [Erratum: Nucl. Phys.B312,751(1989)].

[169] N. Arkani-Hamed and N. Weiner, “LHC Signals for a SuperUnified Theory of Dark Matter”, JHEP12 (2008) 104, arXiv:0810.0714 [hep-ph].

[170] J. Liu, N. Weiner, and W. Xue, “Signals of a Light Dark Force in the Galactic Center”, JHEP 08(2015) 050, arXiv:1412.1485 [hep-ph].

[171] M. Goodsell, J. Jaeckel, J. Redondo, and A. Ringwald, “Naturally Light Hidden Photons inLARGE Volume String Compactifications”, JHEP 11 (2009) 027, arXiv:0909.0515 [hep-ph].

[172] M. Ahlers, J. Jaeckel, J. Redondo, and A. Ringwald, “Probing Hidden Sector Photons throughthe Higgs Window”, Phys. Rev. D78 (2008) 075005, arXiv:0807.4143 [hep-ph].

[173] M. Pospelov, A. Ritz, and M. B. Voloshin, “Bosonic super-WIMPs as keV-scale dark matter”, Phys.Rev. D78 (2008) 115012, arXiv:0807.3279 [hep-ph].

[174] J. Redondo and M. Postma, “Massive hidden photons as lukewarm dark matter”, JCAP 0902(2009) 005, arXiv:0811.0326 [hep-ph].

[175] H. Davoudiasl, H.-S. Lee, and W. J. Marciano, “’Dark’ Z implications for Parity Violation, RareMeson Decays, and Higgs Physics”, Phys. Rev. D85 (2012) 115019, arXiv:1203.2947[hep-ph].

[176] H. Davoudiasl, H.-S. Lee, and W. J. Marciano, “Muon Anomaly and Dark Parity Violation”, Phys.Rev. Lett. 109 (2012) 031802, arXiv:1205.2709 [hep-ph].

[177] M. Pospelov, A. Ritz, and M. B. Voloshin, “Secluded WIMP Dark Matter”, Phys. Lett. B662 (2008)53–61, arXiv:0711.4866 [hep-ph].

[178] L. J. Hall, K. Jedamzik, J. March-Russell, and S. M. West, “Freeze-In Production of FIMP DarkMatter”, JHEP 03 (2010) 080, arXiv:0911.1120 [hep-ph].

[179] M. Raggi, V. Kozhuharov, and P. Valente, “The PADME experiment at LNF”, EPJ Web Conf. 96(2015) 01025, arXiv:1501.01867 [hep-ex].

[180] S. Andreas, C. Niebuhr, and A. Ringwald, “New Limits on Hidden Photons from Past ElectronBeam Dumps”, Phys. Rev. D86 (2012) 095019, arXiv:1209.6083 [hep-ph].

[181] M. Endo, K. Hamaguchi, and G. Mishima, “Constraints on Hidden Photon Models from Electrong-2 and Hydrogen Spectroscopy”, Phys. Rev. D86 (2012) 095029, arXiv:1209.2558 [hep-ph].

[182] H. Davoudiasl, H.-S. Lee, and W. J. Marciano, “Muon g − 2, rare kaon decays, and parityviolation from dark bosons”, Phys. Rev. D89 no. 9, (2014) 095006, arXiv:1402.3620 [hep-ph].

[183] J. D. Bjorken, R. Essig, P. Schuster, and N. Toro, “New Fixed-Target Experiments to Search forDark Gauge Forces”, Phys. Rev. D80 (2009) 075018, arXiv:0906.0580 [hep-ph].

http://dx.doi.org/10.1142/S0217751X14430040




http://dx.doi.org/10.1088/1475-7516/2007/04/010



http://dx.doi.org/10.1088/1475-7516/2011/06/013



http://dx.doi.org/10.1016/0550-3213(88)90266-0

http://dx.doi.org/10.1088/1126-6708/2008/12/104

http://dx.doi.org/10.1088/1126-6708/2008/12/104





http://dx.doi.org/10.1088/1126-6708/2009/11/027







http://dx.doi.org/10.1088/1475-7516/2009/02/005

http://dx.doi.org/10.1088/1475-7516/2009/02/005

























[184] BaBar , J. P. Lees et al., “Search for a Dark Photon in e+ e− Collisions at BaBar”, Phys. Rev. Lett.113 no. 20, (2014) 201801, arXiv:1406.2980 [hep-ex].

[185] A. Anastasi et al., “Limit on the production of a low-mass vector boson in e+e− → Uγ,U → e+e− with the KLOE experiment”, Phys. Lett. B750 (2015) 633–637, arXiv:1509.00740[hep-ex].

[186] A. Bross, M. Crisler, S. H. Pordes, J. Volk, S. Errede, and J. Wrbanek, “A Search for ShortlivedParticles Produced in an Electron Beam Dump”, Phys. Rev. Lett. 67 (1991) 2942–2945.

[187] M. Raggi, “Status of the PADME experiment and review of dark photon searches”, EPJ Web Conf.179 (2018) 01020.

[188] I. Alikhanov and E. A. Paschos, “Searching for new light gauge bosons at e+ e− colliders”, Phys.Rev. D97 no. 11, (2018) 115004, arXiv:1710.10131 [hep-ph].

[189] P. Valente, “Status of positron beams for dark photons experiments”, EPJ Web Conf. 142 (2017)01028.

[190] A. Konaka et al., “Search for Neutral Particles in Electron Beam Dump Experiment”, Phys. Rev.Lett. 57 (1986) 659.

[191] M. Davier and H. Nguyen Ngoc, “An Unambiguous Search for a Light Higgs Boson”, Phys. Lett.B229 (1989) 150–155.

[192] J. Preskill, M. B. Wise, and F. Wilczek, “Cosmology of the Invisible Axion”, Phys. Lett. B120(1983) 127–132.

[193] L. F. Abbott and P. Sikivie, “A Cosmological Bound on the Invisible Axion”, Phys. Lett. B120(1983) 133–136.

[194] M. Dine and W. Fischler, “The Not So Harmless Axion”, Phys. Lett. B120 (1983) 137–141.[195] R. N. Mohapatra and G. Senjanovic, “The Superlight Axion and Neutrino Masses”, Z. Phys. C17

(1983) 53–56.[196] Q. Shafi and F. W. Stecker, “Implications of a Class of Grand Unified Theories for Large Scale

Structure in the Universe”, Phys. Rev. Lett. 53 (1984) 1292.[197] P. Langacker, R. D. Peccei, and T. Yanagida, “Invisible Axions and Light Neutrinos: Are They

Connected?”, Mod. Phys. Lett. 01 (1986) 541.[198] M. Shin, “Light Neutrino Masses and Strong CP Problem”, Phys. Rev. Lett. 59 (1987) 2515.

[Erratum: Phys. Rev. Lett.60,383(1988)].[199] X. G. He and R. R. Volkas, “Models Featuring Spontaneous CP Violation: An Invisible Axion and

Light Neutrino Masses”, Phys. Lett. B208 (1988) 261. [Erratum: Phys. Lett.B218,508(1989)].[200] Z. G. Berezhiani and M. Yu. Khlopov, “Cosmology of Spontaneously Broken Gauge Family

Symmetry”, Z. Phys. C49 (1991) 73–78.[201] S. Bertolini and A. Santamaria, “The Strong CP problem and the solar neutrino puzzle: Are they

related?”, Nucl. Phys. B357 (1991) 222–240.[202] Y. Nomura and J. Thaler, “Dark Matter through the Axion Portal”, Phys. Rev. D79 (2009)

075008, arXiv:0810.5397 [hep-ph].[203] A. De Angelis, M. Roncadelli, and O. Mansutti, “Evidence for a new light spin-zero boson from

cosmological gamma-ray propagation?”, Phys. Rev. D76 (2007) 121301, arXiv:0707.4312[astro-ph].

[204] M. Simet, D. Hooper, and P. D. Serpico, “The Milky Way as a Kiloparsec-Scale Axionscope”, Phys.Rev. D77 (2008) 063001, arXiv:0712.2825 [astro-ph].

[205] M. Sanchez-Conde, D. Paneque, E. Bloom, F. Prada, and A. Dominguez, “Hints of the existenceof Axion-Like-Particles from the gamma-ray spectra of cosmological sources”, Phys.Rev. D79(2009) 123511, arXiv:0905.3270 [astro-ph.CO].

















http://dx.doi.org/10.1016/0370-2693(89)90174-3

http://dx.doi.org/10.1016/0370-2693(89)90174-3

http://dx.doi.org/10.1016/0370-2693(83)90637-8

http://dx.doi.org/10.1016/0370-2693(83)90637-8

http://dx.doi.org/10.1016/0370-2693(83)90638-X

http://dx.doi.org/10.1016/0370-2693(83)90638-X

http://dx.doi.org/10.1016/0370-2693(83)90639-1

http://dx.doi.org/10.1007/BF01577819

http://dx.doi.org/10.1007/BF01577819


http://dx.doi.org/10.1142/S0217732386000683


http://dx.doi.org/10.1016/0370-2693(89)91457-3, 10.1016/0370-2693(88)90427-3

http://dx.doi.org/10.1007/BF01570798

http://dx.doi.org/10.1016/0550-3213(91)90467-C














[206] E. Bulbul, M. Markevitch, A. Foster, R. K. Smith, M. Loewenstein, et al., “Detection of AnUnidentified Emission Line in the Stacked X-ray spectrum of Galaxy Clusters”, Astrophys.J. 789(2014) 13, arXiv:1402.2301 [astro-ph.CO].

[207] A. Boyarsky, O. Ruchayskiy, D. Iakubovskyi, and J. Franse, “Unidentified Line in X-Ray Spectra ofthe Andromeda Galaxy and Perseus Galaxy Cluster”, Phys.Rev.Lett. 113 (2014) 251301,arXiv:1402.4119 [astro-ph.CO].

[208] C. D. R. Carvajal, A. G. Dias, C. C. Nishi, and B. L. Sánchez-Vega, “Axion Like Particles and theInverse Seesaw Mechanism”, JHEP 1505 (2015) 069, arXiv:1503.03502 [hep-ph].

[209] K. N. Abazajian et al., “Light Sterile Neutrinos: A White Paper”, arXiv:1204.5379 [hep-ph].[210] M. Drewes et al., “A White Paper on keV Sterile Neutrino Dark Matter”, JCAP 1701 no. 01,

(2017) 025, arXiv:1602.04816 [hep-ph].[211] S. Dodelson and L. M. Widrow, “Sterile-neutrinos as dark matter”, Phys. Rev. Lett. 72 (1994)

17–20, arXiv:hep-ph/9303287 [hep-ph].[212] P. Sikivie, “Detection Rates for ’Invisible’ Axion Searches”, Phys. Rev. D32 (1985) 2988.

[Erratum: Phys. Rev.D36,974(1987)].[213] P. Sikivie, N. Sullivan, and D. B. Tanner, “Proposal for Axion Dark Matter Detection Using an LC

Circuit”, Phys. Rev. Lett. 112 no. 13, (2014) 131301, arXiv:1310.8545 [hep-ph].[214] Planck , N. Aghanim et al., “Planck 2018 results. VI. Cosmological parameters”,

arXiv:1807.06209 [astro-ph.CO].[215] A. Ringwald and K. Saikawa, “Axion dark matter in the post-inflationary Peccei-Quinn symmetry

breaking scenario”, Phys. Rev. D93 no. 8, (2016) 085031, arXiv:1512.06436 [hep-ph].[Addendum: Phys. Rev.D94,no.4,049908(2016)].

[216] A. G. Dias, V. Pleitez, and M. D. Tonasse, “Naturally light invisible axion and local Z(13) x Z(3)symmetries”, Phys. Rev. D69 (2004) 015007, arXiv:hep-ph/0210172 [hep-ph].

[217] A. G. Dias, V. Pleitez, and M. D. Tonasse, “Naturally light invisible axion in models with largelocal discrete symmetries”, Phys. Rev. D67 (2003) 095008, arXiv:hep-ph/0211107 [hep-ph].

[218] P. Minkowski, “µ→ eγ at a Rate of One Out of 109 Muon Decays?”, Phys. Lett. B67 (1977)421–428.

[219] M. Gell-Mann, P. Ramond, and R. Slansky, “Complex Spinors and Unified Theories”, Conf. Proc.C790927 (1979) 315–321, arXiv:1306.4669 [hep-th].

[220] T. Yanagida, “Horizontal Symmetry and Masses of Neutrinos”, Conf. Proc. C7902131 (1979)95–99.

[221] R. N. Mohapatra and G. Senjanovic, “Neutrino Mass and Spontaneous Parity Violation”, Phys.Rev. Lett. 44 (1980) 912.

[222] J. Schechter and J. W. F. Valle, “Neutrino Masses in SU(2)×U(1) Theories”, Phys. Rev. D22(1980) 2227.

[223] J. Schechter and J. W. F. Valle, “Neutrino Decay and Spontaneous Violation of Lepton Number”,Phys. Rev. D25 (1982) 774.

[224] W. Grimus and L. Lavoura, “The Seesaw mechanism at arbitrary order: Disentangling the smallscale from the large scale”, JHEP 0011 (2000) 042, arXiv:hep-ph/0008179 [hep-ph].

[225] S. M. Boucenna, S. Morisi, and J. W. Valle, “The low-scale approach to neutrino masses”,Adv.High Energy Phys. 2014 (2014) 831598, arXiv:1404.3751 [hep-ph].

[226] H. Hettmansperger, M. Lindner, and W. Rodejohann, “Phenomenological Consequences ofsub-leading Terms in See-Saw Formulas”, JHEP 04 (2011) 123, arXiv:1102.3432 [hep-ph].

[227] P. F. de Salas, D. V. Forero, C. A. Ternes, M. Tortola, and J. W. F. Valle, “Status of neutrinooscillations 2017”, arXiv:1708.01186 [hep-ph].

http://dx.doi.org/10.1088/0004-637X/789/1/13

http://dx.doi.org/10.1088/0004-637X/789/1/13







http://dx.doi.org/10.1088/1475-7516/2017/01/025

http://dx.doi.org/10.1088/1475-7516/2017/01/025















http://dx.doi.org/10.1016/0370-2693(77)90435-X

http://dx.doi.org/10.1016/0370-2693(77)90435-X







http://dx.doi.org/10.1088/1126-6708/2000/11/042


http://dx.doi.org/10.1155/2014/831598






[228] A. Ibarra, E. Molinaro, and S. Petcov, “TeV Scale See-Saw Mechanisms of Neutrino MassGeneration, the Majorana Nature of the Heavy Singlet Neutrinos and (ββ )0ν-Decay”, JHEP1009 (2010) 108, arXiv:1007.2378 [hep-ph].

[229] A. Das and N. Okada, “Inverse seesaw neutrino signatures at the LHC and ILC”, Phys. Rev. D 88(Dec, 2013) 113001. http://link.aps.org/doi/10.1103/PhysRevD.88.113001.

[230] A. Das, P. S. Bhupal Dev, and N. Okada, “Direct bounds on electroweak scale pseudo-Diracneutrinos from

ps = 8 TeV LHC data”, Phys. Lett. B735 (2014) 364–370, arXiv:1405.0177

[hep-ph].[231] CMS , S. Chatrchyan et al., “Searches for long-lived charged particles in pp collisions at

ps=7

and 8 TeV”, JHEP 1307 (2013) 122, arXiv:1305.0491 [hep-ex].[232] M. L. Perl, P. C. Kim, V. Halyo, E. R. Lee, I. T. Lee, D. Loomba, and K. S. Lackner, “The search for

stable, massive, elementary particles”, International Journal of Modern Physics A 16 no. 12,(2001) 2137–2164.

[233] T. Asaka, M. Laine, and M. Shaposhnikov, “Lightest sterile neutrino abundance within thenuMSM”, JHEP 01 (2007) 091, arXiv:hep-ph/0612182 [hep-ph]. [Erratum:JHEP02,028(2015)].

[234] K. Abazajian, G. M. Fuller, and M. Patel, “Sterile neutrino hot, warm, and cold dark matter”,Phys. Rev. D64 (2001) 023501, arXiv:astro-ph/0101524 [astro-ph].

[235] A. Boyarsky, O. Ruchayskiy, and D. Iakubovskyi, “A Lower bound on the mass of Dark Matterparticles”, JCAP 0903 (2009) 005, arXiv:0808.3902 [hep-ph].

[236] A. Boyarsky, J. Lesgourgues, O. Ruchayskiy, and M. Viel, “Lyman-alpha constraints on warm andon warm-plus-cold dark matter models”, JCAP 0905 (2009) 012, arXiv:0812.0010[astro-ph].

[237] S. Horiuchi, P. J. Humphrey, J. Onorbe, K. N. Abazajian, M. Kaplinghat, and S. Garrison-Kimmel,“Sterile neutrino dark matter bounds from galaxies of the Local Group”, Phys. Rev. D89 no. 2,(2014) 025017, arXiv:1311.0282 [astro-ph.CO].

[238] A. Boyarsky, D. Iakubovskyi, and O. Ruchayskiy, “Next decade of sterile neutrino studies”, Phys.Dark Univ. 1 (2012) 136–154, arXiv:1306.4954 [astro-ph.CO].

[239] M. C. Gonzalez-Garcia and J. W. F. Valle, “Enhanced lepton flavor violation with masslessneutrinos: A Study of muon and tau decays”, Mod. Phys. Lett. 07 (1992) 477–488.

[240] F. Deppisch and J. W. F. Valle, “Enhanced lepton flavor violation in the supersymmetric inverseseesaw model”, Phys. Rev. D72 (2005) 036001, arXiv:hep-ph/0406040 [hep-ph].

[241] MEG , J. Adam et al., “New constraint on the existence of the µ+ → e+γ decay”, Phys. Rev. Lett.110 (2013) 201801, arXiv:1303.0754 [hep-ex].

[242] BaBar , B. Aubert et al., “Searches for Lepton Flavor Violation in the Decays τ± → e±γ andτ± → µ±γ”, Phys. Rev. Lett. 104 (2010) 021802, arXiv:0908.2381 [hep-ex].

[243] M. Lindner, M. Platscher, and F. S. Queiroz, “A Call for New Physics : The Muon AnomalousMagnetic Moment and Lepton Flavor Violation”, arXiv:1610.06587 [hep-ph].

[244] A. Das, T. Nomura, H. Okada, and S. Roy, “Generation of radiative neutrino mass in the linearseesaw framework, charged lepton flavor violation and dark matter”, arXiv:1704.02078[hep-ph].

[245] A. Merle, A. Schneider, and M. Totzauer, “Dodelson-Widrow Production of Sterile Neutrino DarkMatter with Non-Trivial Initial Abundance”, JCAP 1604 no. 04, (2016) 003, arXiv:1512.05369[hep-ph].

[246] J. L. Feng, B. Fornal, I. Galon, S. Gardner, J. Smolinsky, T. M. P. Tait, and P. Tanedo, “Particlephysics models for the 17 MeV anomaly in beryllium nuclear decays”, Phys. Rev. D95 no. 3,












http://dx.doi.org/10.1142/S0217751X01003548

http://dx.doi.org/10.1142/S0217751X01003548

http://dx.doi.org/10.1088/1126-6708/2007/01/091




http://dx.doi.org/10.1088/1475-7516/2009/03/005


http://dx.doi.org/10.1088/1475-7516/2009/05/012









http://dx.doi.org/10.1142/S0217732392000434











http://dx.doi.org/10.1088/1475-7516/2016/04/003






(2017) 035017, arXiv:1608.03591 [hep-ph].[247] E. Izaguirre, G. Krnjaic, P. Schuster, and N. Toro, “Testing GeV-Scale Dark Matter with

Fixed-Target Missing Momentum Experiments”, Phys. Rev. D91 no. 9, (2015) 094026,arXiv:1411.1404 [hep-ph].

[248] Y.-S. Liu and G. A. Miller, “Validity of the Weizsäcker-Williams approximation and the analysis ofbeam dump experiments: Production of an axion, a dark photon, or a new axial-vector boson”,Phys. Rev. D96 no. 1, (2017) 016004, arXiv:1705.01633 [hep-ph].

[249] C. F. von Weizsacker, “Radiation emitted in collisions of very fast electrons”, Z. Phys. 88 (1934)612–625.

[250] E. J. Williams, “Correlation of certain collision problems with radiation theory”, Kong. Dan. Vid.Sel. Mat. Fys. Med. 13N4 no. 4, (1935) 1–50.

[251] K. J. Kim and Y.-S. Tsai, “Improved Weizsacker-Williams method and its application to leptonand W boson pair production”, Phys. Rev. D8 (1973) 3109.

[252] S. N. Gninenko, D. V. Kirpichnikov, M. M. Kirsanov, and N. V. Krasnikov, “The exact tree-levelcalculation of the dark photon production in high-energy electron scattering at the CERN SPS”,arXiv:1712.05706 [hep-ph].

[253] NA64 , D. Banerjee et al., “Search for vector mediator of Dark Matter production in invisibledecay mode”, Phys. Rev. D97 no. 7, (2018) 072002, arXiv:1710.00971 [hep-ex].

[254] I. Rachek, D. Nikolenko, and B. Wojtsekhowski, “Status of the experiment for the search of adark photon at VEPP-3”, EPJ Web Conf. 142 (2017) 01025.

[255] J. Alexander, “MMAPS: Missing-Mass A-Prime Search”, EPJ Web Conf. 142 (2017) 01001.[256] M. Raggi and V. Kozhuharov, “Proposal to Search for a Dark Photon in Positron on Target

Collisions at DAΦNE Linac”, Adv. High Energy Phys. 2014 (2014) 959802, arXiv:1403.3041[physics.ins-det].

[257] A. Ghigo, G. Mazzitelli, F. Sannibale, P. Valente, and G. Vignola, “Commissioning of the DAFNEbeam test facility”, Nucl. Instrum. Meth. A515 (2003) 524–542.

[258] PADME , V. Scherini, “Search for the Dark Photon with the PADME Experiment at LNF”, inProceedings, 13th Patras Workshop on Axions, WIMPs and WISPs, (PATRAS 2017): Thessaloniki,Greece, 15 May 2017 - 19, 2017, pp. 227–230. 2018. arXiv:1712.01936 [hep-ex].

[259] H. Bethe and W. Heitler, “On the Stopping of fast particles and on the creation of positiveelectrons”, Proc. Roy. Soc. Lond. A146 (1934) 83–112.

[260] Y.-S. Tsai and V. Whitis, “THICK TARGET BREMSSTRAHLUNG AND TARGET CONSIDERATIONFOR SECONDARY PARTICLE PRODUCTION BY ELECTRONS”, Phys. Rev. 149 (1966)1248–1257.

[261] M. Bohm and W. Hollik, “Radiative Corrections to Polarized e- e+ Annihilation in the StandardElectroweak Model”, Nucl. Phys. B204 (1982) 45–77.

[262] E. Clementi, D. L. Raimondi, and W. P. Reinhardt, “Atomic Screening Constants from SCFFunctions. II. Atoms with 37 to 86 Electrons”, The Journal of Chemical Physics 47 no. 4, (1967)1300–1307, http://dx.doi.org/10.1063/1.1712084.http://dx.doi.org/10.1063/1.1712084.

[263] V. J. Ghosh, M. Alatalo, P. Asoka-Kumar, B. Nielsen, K. G. Lynn, A. C. Kruseman, and P. E.Mijnarends, “Calculation of the Doppler broadening of the electron-positron annihilationradiation in defect-free bulk materials”, Phys. Rev. B 61 (Apr, 2000) 10092–10099.https://link.aps.org/doi/10.1103/PhysRevB.61.10092.

[264] NA48/2 , J. R. Batley et al., “Search for the dark photon in π0 decays”, Phys. Lett. B746 (2015)178–185, arXiv:1504.00607 [hep-ex].








http://dx.doi.org/10.1007/BF01333110

http://dx.doi.org/10.1007/BF01333110







http://dx.doi.org/10.1155/2014/959802



http://dx.doi.org/10.1016/j.nima.2003.07.017

http://dx.doi.org/10.3204/DESY-PROC-2017-02/scherini_viviana


http://dx.doi.org/10.1098/rspa.1934.0140

http://dx.doi.org/10.1103/PhysRev.149.1248

http://dx.doi.org/10.1103/PhysRev.149.1248

http://dx.doi.org/10.1016/0550-3213(82)90421-7

http://dx.doi.org/10.1063/1.1712084

http://dx.doi.org/10.1063/1.1712084

http://arxiv.org/abs/http://dx.doi.org/10.1063/1.1712084

http://dx.doi.org/10.1063/1.1712084

http://dx.doi.org/10.1103/PhysRevB.61.10092

https://link.aps.org/doi/10.1103/PhysRevB.61.10092





[265] J. Blümlein and J. Brunner, “New Exclusion Limits on Dark Gauge Forces from ProtonBremsstrahlung in Beam-Dump Data”, Phys. Lett. B731 (2014) 320–326, arXiv:1311.3870[hep-ph].

[266] J. Blümlein and J. Brunner, “New Exclusion Limits for Dark Gauge Forces from Beam-DumpData”, Phys. Lett. B701 (2011) 155–159, arXiv:1104.2747 [hep-ex].

[267] BaBar , J. P. Lees et al., “Search for Invisible Decays of a Dark Photon Produced in e+ e−

Collisions at BaBar”, Phys. Rev. Lett. 119 no. 13, (2017) 131804, arXiv:1702.03327 [hep-ex].[268] P. deNiverville, C.-Y. Chen, M. Pospelov, and A. Ritz, “Light dark matter in neutrino beams:

production modelling and scattering signatures at MiniBooNE, T2K and SHiP”, Phys. Rev. D95no. 3, (2017) 035006, arXiv:1609.01770 [hep-ph].

[269] S. Hannestad and T. Schwetz, “Cosmology and the neutrino mass ordering”, JCAP 1611 no. 11,(2016) 035, arXiv:1606.04691 [astro-ph.CO].

[270] S. Vagnozzi, E. Giusarma, O. Mena, K. Freese, M. Gerbino, S. Ho, and M. Lattanzi, “Unveiling νsecrets with cosmological data: neutrino masses and mass hierarchy”, arXiv:1701.08172[astro-ph.CO].











http://dx.doi.org/10.1088/1475-7516/2016/11/035

http://dx.doi.org/10.1088/1475-7516/2016/11/035




A G R A D E C I M I E N T O S

Agradezco infinitamente a mi madre por su permanente e incondicionalacompañamiento, por sus bases y sus continuas enseñanzas. Sin ella nada

sería posible...

—

Quiero agradecer a mis orientadores Enrico Nardi y Oscar Zapata por supaciencia y dedicación en el desarrollo de los trabajos obtenidos, por com-partir su conocimiento y enseñarme tantas lecciones que me servirán paracontinuar formándome como persona. Muchas gracias por todo.A mi esposa Yicenia por su constante apoyo, a mis compañeros por todaslas discusiones que sirvieron de una u otra manera en la elaboración deeste proyecto de vida, y a mi familia por todo lo compartido. Agradezcoa la Universidad de Antioquia y al INFN-Laboratori Nazionali di Frascatipor brindarme la oportunidad de formarme académicamente. Finalmente,a COLCIENCIAS por el apoyo económico.

89

C O L O P H O N

This document has been typeset with LATEX using the typographical look-and-feel classicthesis developed by André Miede.All Feynman diagrams have been drawn with Jaxodraw 2.0; data plotshave been plotted using Jupyter notebook.All numerical and symbolical calculations and evaluations have been per-formed with Python and Mathematica.

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