TIFR/TH/18-xxxx

Sterile Neutrinos with Secret Interactions — Cosmological Discord?

Xiaoyong Chu,1, a Basudeb Dasgupta,2, b Mona Dentler,3, c Joachim Kopp,3, 4, d and Ninetta Saviano3, e

1Institute of High Energy Physics, Austrian Academy of Sciences, Nikolsdorfergasse 18, 1050 Vienna, Austria.

2Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai, 400005, India.

3PRISMA Cluster of Excellence and Mainz Institute for Theoretical Physics,

Johannes Gutenberg University, 55099 Mainz, Germany.4Theoretical Physics Department, CERN, 1211 Geneva, Switzerland

Several long-standing anomalies from short-baseline neutrino oscillation experiments – most re-cently corroborated by new data from MiniBooNE – have led to the hypothesis that extra, “sterile”,neutrino species might exist. Models of this type face severe cosmological constraints, and severalideas have been proposed to avoid these constraints. Among the most widely discussed ones aremodels with so-called “secret interactions” in the neutrino sector. In these models, sterile neutrinosare hypothesized to couple to a new interaction, which dynamically suppresses their production inthe early Universe through finite-temperature effects. Recently, it has been argued that the originalcalculations demonstrating the viability of this scenario need to be refined. Here, we update ourearlier results from arXiv:1310.6337 [JCAP 1510 (2015) no.10, 011] accordingly. We confirm thatmuch of the previously open parameter space for secret interactions is in fact ruled out by cosmo-logical constraints on the sum of neutrino masses and on free-streaming of active neutrinos. Wethen discuss possible modifications of the vanilla scenario that would reconcile sterile neutrinos withcosmology.

I. INTRODUCTION

A quote famously attributed to Isaac Asimov is“the most exciting phrase to hear in science, theone that heralds new discoveries, is not ‘Eureka’ but‘That’s funny . . . ’ ”. Neutrino physics is arguably a fieldof research where this phrase can be heard rather fre-quently. Currently, it applies for instance to several in-dependent anomalies observed in short baseline neutrinooscillation experiments [1–6]. Most recently, interest inthese anomalies has been renewed when new data fromthe MiniBooNE experiment at Fermilab corroborated itsearlier results [7]. The anomalies have been interpretedas possible hints for the existence of a fourth (“sterile”)neutrino flavor, even though global fits indicate that it isnot possible to interpret all experimental results in sucha scenario [8–16]. This conclusion remains true in scenar-ios with more than one sterile neutrino [9, 10]. However,the possibility remains that some anomalies are herald-ing new physics while others have mundane explanations.Even more interesting would be the possibility that thenew physics is richer than just a sterile neutrino (seefor instance [17, 18]). In any case, a very severe trialthat sterile neutrino models must face is that of cosmol-ogy. More specifically, observations of the cosmic mi-crowave background (CMB), of light element abundancesfrom Big Bang Nucleosynthesis (BBN), and of large scalestructures (LSS) in the Universe constrain the total en-ergy in relativistic species, usually expressed in terms of

axiaoyong.chu@oeaw.ac.at

bbdasgupta@theory.tifr.res.in

cjmodentle@uni-mainz.de

djkopp@uni-mainz.de

ensaviano@uni-mainz.de

the effective number of neutrino species, Neff [19–21]. Inaddition, LSS and the CMB constrain the sum of neu-trino masses,

∑mν [22–25], or, more precisely, the sum

of the masses of collisionless neutrino species.

However, cosmology can only probe particle speciesthat are abundant in the early Universe. It is thereforeinteresting to explore scenarios where sterile neutrinos, inspite of having O(10%) mixing with the active neutrinos,are not produced in sufficient abundance to have observ-able consequences. One proposed mechanism to achievethis is the “secret interactions” scenario [26, 27], in whichsterile neutrinos, while being singlets under the SM gaugegroup, are coupled to a new U(1)′ gauge boson A′ (orto a new pseudoscalar [28–30]) with mass M � MW .Through this new interaction, sterile neutrinos feel anew temperature-dependent matter potential, which dy-namically suppresses their mixing with active neutrinosat high temperatures, while being negligible today. Toavoid constraints on Neff, it is in particular required thatactive–sterile neutrino mixing is strongly suppressed attemperatures & MeV, the temperature where active neu-trinos decouple from the photon bath. Note that intro-ducing new interactions in the sterile neutrino sector mayalso be one way of reconciling the LSND and MiniBooNEanomalies with other neutrino oscillation data [17].

While the secret interactions scenario has motivateda number of model building and phenomenology pa-pers [28, 30–43], it has also been argued that most of theavailable parameter space is ruled out. Constraints comemainly from two directions. First, sterile neutrinos willeventually recouple with active neutrinos and are then ef-ficiently produced collisionally via the Dodelson–Widrowmechanism [35, 44]. The temperature at which this re-coupling happens depends on the interplay of the effectivepotential that suppresses flavor-changing collisions andthe relevant scattering rates, which can be very large (see

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below) [41]. Even if recoupling happens at T � MeV,it will still lead to equilibration between active and ster-ile neutrinos. This may lead to tension with limits on∑mν [36]. Second, mixing of active and sterile neutri-

nos leads to reduced free-streaming of active neutrinos. Acertain amount of active neutrino free-streaming is, how-ever, required by CMB observations [31, 42]. Note thatthis second constraint spoils attempts to keep sterile neu-trinos safe from the limit on

∑mν by postulating that

they interact so strongly that they cannot free-streamenough to affect large scale structure [40].

In this paper, we take two more steps in the ongo-ing exploration of secretly interacting sterile neutrinos.First, we update our earlier results from ref. [40], con-firming in particular the findings by Cherry et al. [41].A detailed account of cosmological constraints on secretinteractions has also been given recently in [43]. In com-parison to that paper, we focus less on a complete fit tocosmological data based on simulations, but instead de-rive constraints from physical arguments and estimatesthat can be much more easily generalized to other mod-els. Second, and perhaps more importantly, we show thatalthough the vanilla secret interactions model is indeeddisfavored by cosmological data, the general idea under-lying it remains viable and interesting. We give explicitexamples of models that show this. The core assump-tion of the secret interactions scenario is that the sterileneutrino is hidden in cosmology because it gets a largetemperature-dependent mass at high T due to its inter-actions. We show that, if the vector boson employed inthe original works [26, 27] is replaced by a scalar medi-ator with suitable symmetries and potential, the above-mentioned constraints from BBN, CMB and LSS appearto be avoidable. The mechanism we propose generatesa large mass for νs in the high-temperature phase of thescalar potential, precluding efficient νs production. Onlyafter a late phase transition in the scalar sector is thesterile neutrino mass reduced to the value observed to-day. We also outline more mundane ways to reconcile thevanilla scenario with data, by simply adding more free-streaming particles, or by allowing neutrinos to decay.

We begin with a review of the basic features of thesecret interactions scenarios (section II), followed bythe derivation of detailed cosmological constraints (sec-tion III). We then discuss several possible modificationsto the original secret interactions models that could ren-der the scenario phenomenologically viable again (sec-tion IV). We summarize and conclude in section V.

II. THE SECRET INTERACTIONS SCENARIO

We augment the Standard Model (SM) with an extra,sterile, neutrino flavor νs. We assume νs has appreciable,O(10%), mixing with the three active neutrino flavors,collectively denoted by νa. For the neutrino mass eigen-states, we use the notation νj , with j = 1 . . . 4, where ν1,ν2, ν3 have masses � 1 eV and are mostly composed of

νa. For the mostly-sterile mass eigenstate ν4, we assumea mass around 1 eV, as motivated by the short baselineoscillation anomalies [13, 14, 16]. We finally introducethe secret interaction by charging the sterile flavor eigen-state νs under a new U(1)s gauge group, with a gaugeboson A′ at the MeV scale or somewhat below. The rel-evant interaction term reads

Lint = esν̄sγµPLνsA

′µ , (1)

where es is the U(1)s coupling constant, and

PL =12 (1− γ

5) is the projection operator onto left-chiralfermion states. In the following, we will be agnostic aboutthe mechanism that breaks U(1)s and endows the A

′ bo-son with a mass. In particular, we will neglect the pos-sible additional degrees of freedom—for instance sterilesector Higgs bosons—that may be introduced to achievethis breaking. If A′ gets its mass M via the Stückelbergmechanism, this approximation becomes exact. When asterile neutrino with energy E propagates through a ther-malized background of sterile neutrinos and A′ bosons attemperature Ts, it experiences a potential [27]

Veff '

−7π

2e2sET4s

45M4for Ts �M

+e2sT

2s

8Efor Ts �M

. (2)

Like a conventional Mikheyev–Smirnov–Wolfenstein(MSW) potential [45–47], Veff changes the neutrino mix-ing angle. In the 1 + 1 flavor approximation, the νs–νamixing angle θm in a thermal νs background is given by

sin2 2θm =sin2 2θ0(

cos 2θ0 +2E

∆m2Veff

)2+ sin2 2θ0

, (3)

where θ0 is the mixing angle in vacuum, and ∆m2 ≡

m24 − m21 is the mass squared difference between the

mostly sterile and mostly active neutrino mass eigen-states. Our qualitative results will remain unchangedeven when more than one active neutrino flavor is con-sidered. Equations (2) and (3) show that, at the hightemperatures prevalent in the early Universe, mixing isstrongly suppressed because |Veff| � ∆m

2/(2E). Exper-iments today, on the other hand, will observe θm = θ0 toa very good approximation.

Note that Veff has opposite sign at Ts �M comparedto Ts �M . This implies that there should be a temper-ature range around Ts ∼ M where the potential passesthrough zero and has a small magnitude. In other words,sterile neutrinos could be produced in this interval. How-ever, as shown explicitly in ref. [27], this temperature in-terval is very short, therefore it is unclear what its impacton the final νs abundance is. Answering this question isone of the goals of this paper.

A typical cosmological history in the secret interac-tions model begins at high temperature with a negligible

3

abundance of νs and A′. As soon as a small number of

νs and A′ are produced through oscillations or through

some high-scale interactions, a large effective potentialVeff arises, suppressing mixing and preventing further νsproduction. When the temperature drops so low that|Veff| . ∆m

2/(2E), sterile neutrinos recouple and ther-malize with νa through unsuppressed oscillations and A

′-mediated scattering processes. At |Veff| ' ∆m

2/(2E),oscillations can even be resonantly enhanced if the re-coupling temperature is . M so that Veff is negative. Ifthis recoupling between νa and νs happens after the νahave decoupled from the thermal bath at temperatures∼ MeV, νs production does not change Neff. The pre-dicted value of Neff is then similar to that in the SM,Neff = 3.046 [48, 49], and the model is consistent withthe observed value Neff = 3.15±0.23 (68% CL) [21]. (Thevalue of Neff measured at recombination can still be re-duced compared to the SM prediction because νs becomenon-relativistic earlier than νa.) Nevertheless, the con-version of νa into νs increases the prediction for

∑mν ,

and for eV-scale νs, this puts the model in tension withthe constraint

∑mν < 0.23 eV [21]. Note that, to be

precise, these bounds should be slightly modified in thesecret interactions scenario as νa–νs recoupling at sub-MeV temperatures lowers the temperature of the neu-trino sector compared to the standard ΛCDM model [36].Computing this correction would require modified simu-lations of structure formation, which is beyond the scopeof this work.

In ref. [40], we had proposed two possible ways out:

(i) Recoupling between νa and νs never happens be-cause the gauge coupling es is so small that the ster-ile neutrino scattering rate Γs drops below the Hubblerate before |Veff| drops below ∆m

2/(2E). Of course,es still needs to be large enough to make sure that|Veff| � ∆m

2/(2E) until active neutrino scattering de-couples. However, using a refined calculation, that wewill confirm below in section III, the authors of ref. [41]have argued that these two contrary requirements cannotbe fulfilled simultaneously.

(ii) Recoupling between νa and νs happens, but thegauge coupling es is so large that νs cannot free-streamuntil very late times, after matter–radiation equality. Inthis case, bounds on

∑mν , which are effectively bounds

on free-streaming species, do not apply. A possible prob-lem with this option is that the free-streaming of νa willbe delayed as well through the νa–νs mixing. The roughestimates given in [40], suggested that the scenario mightbe marginally consistent with the data and only a ded-icated analysis of CMB data would allow us to drawdefinitive conclusions. Forastieri et al. have recently car-ried out such an analysis and have shown that the sce-nario appears to be in tension with data [42].

III. CONSTRAINTS ON STERILE NEUTRINOSWITH SECRET INTERACTIONS

In the following, we will derive updated constraintson the secret interactions model introduced in section II.We employ two complementary approaches: our first ap-proach, outlined in section III A, is a computation ofthe recoupling temperature, Trec, i.e., the temperatureat which the sterile and active neutrinos recouple in the1+1 scenario. This is similar to our previous calculationsin ref. [40], but includes several improvements includingthose pointed out in ref. [41]. Assuming that sterile andactive neutrinos equilibrate instantaneously at Trec, thiscomputation allows us to estimate which cosmologicaldata sets are sensitive to the resulting abundance of ster-ile neutrinos. In the second approach, presented in sec-tion III B, we go one step further and explicitly simulatethe flavor evolution of the neutrino sector after recou-pling. In doing so, we also go beyond the 1 + 1 flavorapproximation and use instead a 2 + 1 flavor approxima-tion, i.e., two active flavors and one sterile flavor. Thereare several motivations to do this, as we will explain later.

A. Recoupling Temperature Computation

In our first approach, we work in the mass basis andcompute the production rate Γs of the mostly sterile masseigenstate. The following reactions contribute to νs ≈ ν4production:

1. W and Z-mediated processes

(i) e− + e+ → ν̄1 + ν4 via s-channel Z exchangeor t-channel W exchange;

(ii) e− + ν1 → e− + ν4 via t-channel Z exchange

or s-channel W exchange;

(iii) e+ + ν1 → e+ + ν4 via t-channel Z exchange

or s-channel W exchange;

(iv) ν1 + ν1 → ν1 + ν4 via Z exchange in the t- oru-channel;

(v) ν̄1 + ν1 → ν̄1 + ν4 via Z exchange in the s- ort-channel;

2. A′-mediated processes

(vi) ν̄4 + ν1 → ν̄4 + ν4 via A′ exchange in the s- or

t-channel;

(vii) ν4 + ν1 → ν4 + ν4 via A′ exchange in the t- or

u-channel;

Of course, the corresponding CP -conjugate processescontribute equally. Analytical expressions for the crosssections of these reactions are given in the appendix. Ofthe processes listed here, the first five are SM reactions in-volving electrons and/or electron neutrinos that producesterile neutrinos through the mixing of the light neutrinoswith the heavy mass eigenstate ν4. The remaining two

4

processes are mediated by A′ and produce sterile neutri-nos from electron neutrinos through the overlap of νe andν4. Note that process (vi), ν̄4 + ν1 → ν̄4 + ν4, involvess-channel A′ exchange and is thus resonantly enhancedin a specific part of the neutrino spectrum. Note alsothat A′-mediated t-channel scattering is enhanced in theforward direction if the A′ mass is much smaller than theneutrino temperature.

We first compute the temperature Trec at which sterileand active neutrinos recouple via scattering. We defineTrec as the temperature at which Γs becomes equal tothe Hubble rate, i.e., Γs = H. In terms of the scatteringcross sections given in the appendix A, Γs is given by

Γs = cQZ

[〈σv〉ee→14

n2enν

+ 〈σv〉e1→e4 ne

+ 〈σv〉11→14 nν + 〈σv〉14→44 ns], (4)

Here, the notation 〈·〉 refers to averaging over the momen-tum distributions of the involved particles. We assumethese distribution to have a Fermi-Dirac form at all times.Note also that all cross sections depend on the the ster-ile sector temperature Ts through the mixing angle θm.The shorthand notation 〈σv〉ee→14 refers to process (i)above, 〈σv〉e1→e4 refers to the sum of processes (ii) and(iii), 〈σv〉11→14 to the sum of processes (iv) (multipliedby a factor 1/2 to account for the identical particles inthe initial state) and (v), and 〈σv〉14→44 to the sum ofprocesses (vi) and (vii). The factors ne, nν , and ns arethe electron, active neutrino, and sterile neutrino numberdensities, respectively, not including their anti-particles.They are chosen such that each term in eq. (4) gives theproduction rate per active neutrino, i.e., the number ofsterile neutrinos produced per unit time in a spatial vol-ume element occupied on average by one active neutrino.The prefactor cQZ accounts for the quantum Zeno effect,i.e., for the suppression of νs production when the scat-tering rate is faster than the oscillation frequency [50].In this case, oscillations have no time to develop beforethey are interrupted by scattering. To account for thiseffect, we define cQZ as

cQZ =(Lscat/Losc)2

1 + (Lscat/Losc)2, (5)

where Lscat is the νs–νs scattering length and Losc is the

oscillation length in medium. With this definition, cQZis close to one when Lscat � Losc and approaches zerowhen Lscat � Losc.

B. Multi-flavor evolution

To understand the dynamics of sterile neutrino pro-duction in more detail, we have also simulated the evolu-tion of a 2 + 1 system (two active species and one sterile

species) numerically. We do so, (i) to verify that ther-malization between active and sterile neutrinos is indeedquasi-instantaneous after recoupling, (ii) to assess theimpact of a nearly vanishing Veff at Ts ∼ M , (iii) tocheck that the simplified treatment of the quantum Zenocorrection in section III A is valid, and (iv) to investi-gate the possible impact of going beyond the two-flavorapproximation.

As a complete numerical simulation of the flavor evolu-tion including the exact temperature-dependence of Veffis numerically highly challenging, we focus on the evo-lution during the epochs where the effective potential issmall compared to the vacuum oscillation frequency, sothat sterile neutrino mixing is unsuppressed. We use theexact temperature-dependence of Veff from ref. [27] to de-termine the relevant temperature intervals, and then sim-ulate the flavor evolution within these intervals, settingVeff = 0. Our simulation code is based on refs. [35, 36, 42].

The effective potential for a sterile neutrino with 4-momentum k is given by [27]

Veff = −1

2~k2

[[(k0)2 − ~k2]tr

(/uΣ(k)

)− k0tr

(/kΣ(k)

)],

(6)

with Σ(k) the temperature-dependent sterile neutrinoself energy at one-loop and u = (1, 0, 0, 0) the 4-momentum of the heat bath. We use the ultra-relativisticapproximation k0 ' |~k| + Veff and expand eq. (6) inVeff. We can then solve numerically for the critical pointswhere the condition |Veff| = ∆m

2/(2E) is fulfilled.

At high enough temperatures, |Veff| always exceeds∆m2/(2E) as long as the fine structure constant αs (≡e2s/4π) is not zero, but as temperatures become smallertwo possibilities present themselves. The first possibilityis that once |Veff| falls below ∆m

2/(2E), it never exceedsit again. An example of this is shown in the left panel offig. 1. The second possibility is that Veff crosses throughzero but then takes large negative values so that |Veff|exceeds ∆m2/(2E) again, as shown in the right panel offig. 1. We refer to the temperature at which |Veff| in-tersects the vacuum term for the last time as the “lastcrossing” temperature. In the second scenario, |Veff| in-tersects the vacuum term around the zero crossing as well(fig. 1 right), and we call the corresponding temperatureinterval the “zero-crossing” interval.

We describe the neutrino ensembles in terms of amomentum-integrated 3× 3 matrix of densities,

ρ =

ρee ρeµ ρesρµe ρµµ ρµsρse ρsµ ρss

, (7)and a similar expression for antineutrinos, denoted byρ̄. The diagonal entries are the respective number densi-ties, while the off-diagonal ones encode phase informationand vanish for zero mixing. In the standard situation,the equilibrium initial condition for the active neutrino

5

Figure 1. Absolute value of the effective potential Veff as a function of the (active) neutrino temperature Tν , for two different

choices of the mediator mass M and the secret fine structure constant αs ≡ e2s/(4π). Positive (negative) values of Veff are

indicated by solid (dashed) lines. The vacuum oscillation frequency ∆m2/(2E) is displayed as a black line. The temperatures

at which |Veff| and the vacuum oscillation frequency intersect are highlighted in red. We refer to the temperature of the last(left-most) intersection as the “last crossing” temperature. For some choices of M and αs, intersections between |Veff| and∆m

2/(2E) also occur around the temperature where Veff changes sign, as can be observed in the right panel. In this case, we

refer to the short time interval in which |Veff| < ∆m2/(2E) as the “zero-crossing” interval. In our simulations, we assume no

sterile neutrino production when |Veff| > ∆m2/(2E), and we set Veff = 0 whenever |Veff| > ∆m

2/(2E).

number densities is ρee = ρµµ = 1 (and similarly for ρ̄),while for the sterile species we have the initial conditionρss = ρ̄ss ' 0. The normalization of ρ and ρ̄ is chosensuch that a diagonal entry of 1 corresponds to the abun-dance of a single neutrino (or antineutrino) species in theStandard Model.

The evolution equation for ρ is [51–53]

idρ

dt= [Ω, ρ] + C[ρ] . (8)

Once again, a similar equation holds for the antineutrinodensity matrix ρ̄. Here, t is the comoving observer’sproper time. The evolution can be easily recast into afunction of the photon temperature Tγ . The first termon the right-hand side of eq. (8) describes flavor oscilla-tions, with the Hamiltonian given by

Ω = U†〈

m2ν2pU

〉+√

2GF

[− 8〈p〉

3

(E`

M2W+

Eν

M2Z

)],

(9)

where mν = diag(m1,m2,m4) is the neutrino mass ma-trix in the mass basis, and U is the 3 × 3 neutrino mix-ing matrix. The latter depends on three mixing angles,θeµ, θes, and θµs, using the same parameterization asin ref. [53]. We take θeµ equal to the active neutrinomixing angle θ13 [54], and we fix the active–sterile mix-ing angles and mass-squared differences at the best-fitvalues obtained from a global fit to the short-baselineanomalies [16]. The terms proportional to the Fermi

constant GF in eq. (9) encode SM matter effects in neu-trino oscillations. In particular, the term containing E`describes charged current interactions of neutrinos withthe isotropic background medium, related to the energydensity (∝ T 4γ ) of e

± pairs. The term containing Eνdescribes instead interactions of neutrinos with them-selves (self-interaction term), related to the energy den-

sity (∝ (% + %̄)T 4ν ) of ν and ν̄. Note that in both terms,it is necessary to go beyond the low energy effective fieldtheory of SM weak interactions (Fermi theory) and takeinto account momentum-dependent corrections. Thesecorrection terms can compete with the leading term frompure Fermi theory because the latter is proportional tothe tiny lepton asymmetry of the Universe. Further de-tails are given in ref. [53]. We remind the reader that weset Veff = 0 below the last crossing temperature and dur-ing the zero-crossing interval. Moreover, we also neglectthe small neutrino–antineutrino asymmetry ∝ (%− %̄)T 3ν .

The second term on the right-hand side of eq. (8) isthe collisional term. It receives contributions from bothSM and secret interactions:

C[ρ] = CSM[ρ] + CA′ [ρ] . (10)

Following [53], we write the SM collision term as

CSM[ρ] = −i

2G2F

[{S2, ρ− 1} − 2S(ρ− 1)S

+ {A2, ρ− 1}+ 2A(ρ̄− 1)A], (11)

6

12.0 12.2

0.

0.2

0.4

0.6

0.8

1.

Tν [MeV]

Ts,ini=0.3Tγ

αs = 1.× 10-12, M = 10 MeV

ρss

ρμμρ

ee

0.76 0.82

0.

0.2

0.4

0.6

0.8

1.

Tν [MeV]

Ts,ini=0.3Tγ

αs = 1.× 10-9, M = 1 MeV

ρss

ρμμ

ρ ee

0.7965 0.7966

0.

0.2

0.4

0.6

0.8

1.

Tν [MeV]

Ts,ini=0.3Tγ

αs = 1.× 10-6, M = 1 MeV

ρss

ρμμ

ρee

(a) (b) (c)

Figure 2. Evolution of the neutrino density matrix as a function of the (SM) neutrino temperature. We show the temperaturedependence of the νe abundance (ρee), the νµ abundance (ρµµ), and the νs abundance (ρss) in the 2+1 scenario for threedifferent parameter points as indicated in the plots. The gray bands delimit the temperature ranges in which the system isevolved numerically. Panel (a) corresponds to evolution beyond the last crossing temperature, panels (b) and (c) correspondto evolution within the zero-crossing interval. See text for details.

where the matrices S and A contain the numerical co-efficients for the scattering and annihilation of the dif-ferent flavors. In flavor space, they are given by S =diag(ges , g

µs , 0) and A = diag(g

ea, g

µa , 0). Numerically one

finds [55, 56]: (ges)2 = 3.06, (gµs )

2 = 2.22, (gea)2 = 0.5,

(gµa )2 = 0.28.

The collision term corresponding to secret interactionsin the sterile sector can be written schematically as

CA′ [ρ] = −1

2(Γno-res + Γres)

×[{S2A′ , ρ− 1} − 2SA′(ρ− 1)SA′

], (12)

with the coefficient matrix SA′ ≡ diag(0, 0, 1) [35]. Notethat here, we write the scattering processes in the flavorbasis, whereas in section III A we had worked in the massbasis. Thus, the processes contributing to CA′ are νsνs →νsνs, νsν̄s → νsν̄s, and νsν̄s → A

′A′. The coupling to νeis generated by the oscillation terms in the equations ofmotion, but not explicitly present in CA′ . After all, thenew interaction couples only to sterile neutrinos.

To highlight the qualitative differences between dif-ferent contributions to the collision term, we have ar-tificially split CA′ into a piece containing non-resonantscattering processes (including t-channel processes) anda piece containing the contribution from resonantly en-hanced νsν̄s → νsν̄s scattering through s-channel A

′ ex-change. The former piece contains the scattering rate

Γno-res '16π2α2sT

5ν

T 2νM2 +M4

, (13)

while the latter one contains

Γres 'M

Tν· nress · σCM · v ' 6× 10

−2αsM2

Tν. (14)

Note that, at T �M , Γres would receive an extra Boltz-mann suppression factor. In these expressions, we omitO(1) numerical prefactors for simplicity. We use the no-tation nress = 0.06TνMΓA′ for the number density of neu-trinos participating in the resonant s-channel process, i.e.particles whose energies fall within the resonance windowof width ΓA′ = αsM/3π (in the center of mass frame);

σCM = π/M2 is the s-channel cross section in the center

of mass frame, and v = M/Tν is the relative velocity ofthe two neutrinos forming a resonant pair.

In fig. 2 we show the results of the numerical flavorevolution for sterile and active neutrinos at three repre-sentative parameter points. In particular, we show theevolution of the density matrix components ρee (elec-tron neutrino abundance relative to a fully thermalizedspecies), ρµµ (muon neutrino abundance), and ρss (ster-ile neutrino abundance). Panel (a), where M = 10 MeV,

αs = 10−12 was assumed, corresponds to the evolution

after the last crossing temperature, while panel (b), with

M = 1 MeV, αs = 10−9, and panel (c), withM = 1 MeV,

αs = 10−6, correspond to the evolution during the zero

crossing interval. In computing the zero-crossing tem-peratures and the last crossing temperature, we need tomake an assumption on the initial temperature Ts,ini, be-fore any νs are produced via oscillations. Here, we as-sume Ts,ini = 0.3Tγ at Tγ = 1 TeV. We will motivate thischoice below in section III C. Our assumptions on Ts,iniand on the evolution of Ts are of course irrelevant to theactual numerical evolution of the neutrino ensemble as

7

we set Veff = 0 in the zero-crossing interval and afterthe last crossing temperature. The grey bands in fig. 2delimit the temperature range during which we performthe numerical flavor evolution. Within the gray bands,Veff cannot be considered negligible any more. In thecases shown in panels (a) and (b), sterile neutrinos arecopiously produced. In particular, in panel (a), sterileneutrinos are fully thermalized (ρss = 1) by oscillationswith active neutrinos occurring at temperatures around10 MeV and increasing the number of relativistic degreesof freedom Neff. In panel (b) instead, since oscillationsand thus sterile neutrino production happen after theneutrino sector has decoupled from the photon bath, thetotal neutrino number density remains constant. Conse-quently, the asymptotic values of ρee, ρµµ, and ρss areall 0.67 in the 2 + 1 scenario. At the parameter pointsshown in panels (a) and (b), the oscillation rate is muchlarger than the νs scattering rate, i.e., there is no quan-tum Zeno suppression, and the νs scattering rate is inturn much larger than the Hubble rate, i.e., scattering-induced production is efficient. In panel (c) of fig. 2, νsare not copiously produced during the zero-crossing in-terval. The reason is that resonant scattering mediatedby an s-channel A′ is faster than vacuum oscillations sothat νs production is quantum Zeno-suppressed. Notethat this is not the typical behavior – for most combina-tions of M and αs, we find efficient νs production duringthe zero-crossing interval, except for a few cases wherethe interval is very short and/or the mediator is verymassive (∼ 1 GeV). We always find efficient νs produc-tion below the last crossing temperature.

C. Results

In fig. 3 we show the main constraints on the param-eter space of the secret interactions model in the planespanned by the secret gauge boson mass M and the cor-responding fine structure constant αs. We have assumeda sterile neutrino mass ms = 1 eV, and a vacuum mixingangle θ0 = 0.1. We distinguish three regimes: in thevertically striped (blue/orange) region labeled “

∑mν

too large”, νs production is efficiently suppressed downto temperatures Tγ ≤ 1 MeV, so that Neff limits areevaded. Nevertheless, νs are efficiently produced via col-lisional decoherence at late times [36], i.e., around orbelow the last crossing temperature, so that the con-straint on

∑mν is violated. Note that for lighter sterile

neutrinos, ms . 0.2 eV, these parameter regions wouldbe experimentally allowed. The dashed line within thevertically hatched region indicates where the recouplingtemperature equals the A′ mass. In the cross-hatched(brown) region in fig. 3, sterile neutrinos recouple aboveTγ ∼ 1 MeV. They can thus fully thermalize with the SMthermal bath, and as a consequence violate constraintson both Neff and

∑mν . The red shaded region at the

top left of the plots is likely ruled out by CMB data be-cause of insufficient active neutrino free-streaming [42].

The red stars are two benchmark points considered inref. [40]. We see that both are now disfavored. Theboundary between the striped and cross-hatched regionsis first based on the value of Trec calculated using themethods from section III A. These methods, however, donot properly take into account νs production during theshort time interval where zero-crossing happens and afterthe last crossing. Therefore, we use the numerical simu-lations from section III B to reexamine the zero crossinginterval and to determine whether νs production aroundthe zero crossing shifts Trec to larger values. If so, weset Trec to the central temperature of the zero crossinginterval. We find, however, that this correction never af-fects the boundary between the striped and cross-hatchedregions in fig. 3. We conclude that the sum of neutrinomasses constraint and active neutrino free-streaming con-straint together rule out all of the parameter space for themodel [42].

The left and middle panels in fig. 3 correspond to dif-ferent choices of the initial temperature Ts,ini of νs and

A′ at very early times. (We arbitrarily define Ts,ini asthe value of Ts at photon temperature Tγ = 1 TeV.)We assume that there exist some additional new inter-actions between νs and SM particles (for instance in thecontext of a Grand Unified Theory) that lead to ther-malization of νs at a very high temperature Tγ � TeV.When these interactions freeze out (still at Tγ � TeV),the sterile and SM sectors decouple. Afterwards, Ts andTγ may drift apart, and the amount by which they doso above Tγ = 1 TeV is encoded in our choice of Ts,ini.Of course, further entropy is produced in the SM sec-tor at Tγ < 1 TeV, which implies that Ts and Tγ willdrift further apart as the Universe evolves. This effect istaken into account in our calculations. Even if the sterileneutrino abundance is zero after inflation and reheating,and sterile neutrinos are only produced via oscillations,a non-vanishing Ts,ini is still determined by the equationΓs = H. In other words, at any given epoch sterile neu-trinos will be produced until Veff becomes large enough toshut production off (or until full thermal equilibrium be-tween νs and νa is reached). In the right panel of fig. 3, weshow also constraints under the hypothesis that Ts = Tν ,i.e. that the sterile sector temperature follows the activeneutrino temperature at all times. This scenario, whiledifficult to realize in a consistent model, can be consid-ered an upper limit on Ts.

Among the various νs production processes listedabove, the W - and Z-mediated ones are dominant at therecoupling time if either A′ is heavy (close to 1 GeV), or

for αs . 10−14, as shown by the gray region in fig. 4.

The A′-mediated s-channel contribution to process (vi),ν̄4 + ν1 → ν̄4 + ν4 (shown in red), is dominant for mostof the parameter region shown in fig. 4, largely due tothe on-shell resonance. The A′-mediated t-channel con-tributions to processes (vi) (ν̄4 + ν1 → ν̄4 + ν4) and (vii)(ν4 + ν1 → ν4 + ν4), shown in blue, become more impor-tant when either the s-channel resonance is Boltzmann-suppressed in the case of heavy A′, or when the forward

8

Figure 3. The parameter space of the secret interactions model as a function of the secret gauge boson mass M and thecorresponding fine structure constant αs, for three different assumptions on the ratio of the sterile and active sector temperatures(see text for details). We have assumed a sterile neutrino mass ms = 1 eV, and a vacuum mixing angle θ0 = 0.1. The cross-hatched region (brown) is ruled out because it leads to recoupling at Tγ > 1 MeV, so that sterile neutrinos will fully thermalizewith the SM plasma, in violation of the constraints on Neff and

∑mν . In the vertically striped (blue/orange) region, recoupling

occurs at Tγ < 1 MeV, but even in this case νs are produced collisionally after |Veff| drops below ∆m2/(2E), and the model

is ruled out due to constraints on∑mν [36]. This constraint would be avoided for ms . 0.2 eV. The color gradient in this

region, from dark orange to blue, represents the increasing recoupling temperature from 0.05 MeV to 1 MeV. The red shadedregion in the top left of the figure is ruled out because of insufficient active neutrino free-streaming [42]. The red stars indicatetwo benchmark points that were considered in ref. [40] and are now ruled out. There is no parameter region where recouplingnever occurs, i.e., all values of M and αs are ruled out for ms ≥ 1 eV.

enhancement of the t-channel diagrams becomes signifi-cant in the case of very light A′.

Note that the A′ resonance in the s-channel is respon-sible for ruling out the parameter region in which it waspreviously thought [40] that no recoupling between νaand νs happens. The calculations in [40] were based onnaive dimensional arguments, and the enhancements in-duced by on-shell resonance and forward scattering weremissed. This subtle issue was pointed out by Cherry etal. [41], who in particular explained that while the trulyforward scattering of νs only gives a refractive index Veff(which was included in previous papers), multiple “al-most forward” small-angle scatterings, which were incor-rectly ignored, eventually add up to give large angle scat-tering and spatially separate the ν1 and ν4 eigenstatescausing decoherence.

Let us finally address the potential loopholes in ourline of argument so far. We have already argued thatour results – in particular fig. 3 – are unaffected by ster-ile neutrino production during the zero crossing inter-val. We have explicitly checked this using the simula-tions described in section III B. Similarly, the robustnessof our results with regard to possible corrections from themore detailed simulations also addresses the other pointsraised at the beginning of section III B. In particular, itillustrates that the approximation of quasi-instantaneousthermalization after after Veff drops below ∆m

2/(2E) isa good one, that a simplified treatment of the quantumZeno effect is usually justified, and that there are no qual-itative differences between the 1 + 1 and 2 + 1 scenarios.

��-� ��-� ��-� ��� ��� ��� �����-����-����-����-����-���-���-���-����

������ �������� ���� � [���]

����������

�����������������

α � �������� ������� �� ���� (��� ���=�γ)�-���

������

�-������� ��

�� �/�

Figure 4. Dominant scattering channel for collisional νs pro-duction at Trec as a function of M and αs. We have chosenms = 1 eV, and Ts,ini = Tγ . The mixing angle suppression iscommon to all processes.

IV. RECONCILING SECRET INTERACTIONSWITH COSMOLOGY

While fig. 3 shows that the secret interactions model inits vanilla form is difficult to reconcile with cosmologicalconstraints, it is interesting to ask what modifications arenecessary to render it viable. In the following we outlinesome ideas for modifying the scenario in order to reconcile

9

eV-scale sterile neutrinos with cosmology. Some of theideas discussed here may seem rather contrived, but wediscuss them nevertheless to give the reader a feeling forwhat it takes to make eV-scale νs cosmologically viable.

We are not going to comment here on ways of reconcil-ing sterile neutrinos with cosmology that do not involvesecret interactions.

A. Recoupling never happens

If sterile neutrinos are much heavier than the sterilesector temperature Ts at early times and only becomelight when νs-producing processes have decoupled, theirproduction will be suppressed.

One way to implement this idea is to use a scalar me-diator φ instead of a vector mediator A′. The resultingYukawa coupling φν̄sνs makes the mass of νs dependenton the vacuum expectation value (vev) vφ of φ. If vφ islarge at high temperatures and vanishes at lower tem-peratures, νs can be “hidden” until the Universe is coldenough to suppress their production. Scenarios of thistype have been known for a long time [57–69].

To construct a toy model based on this idea, we con-sider two real scalars φ1, φ2, enjoying a Z2×Z2 symmetryunder which φ1 carries charges (−,+), while φ2 carriescharges (+,−). The tree level scalar potential is then

V =λ14φ41 +

λ24φ42 +

λp2φ21φ

22 +

µ212φ21 +

µ222φ22 . (15)

Boundedness of the potential from below requires thatλ1,2 > 0 and λ

2p < λ1λ2. As long as µ

21 > 0 and µ

22 > 0,

there are no broken symmetries at zero temperature.At high temperatures the potential receives thermal

corrections. At 1-loop order and with T 2s � µ21,2, these

are [57]

∆V (Ts) =T 2s24

∑i=1,2

∂2V

∂φ2i

' T2s

24

[(3λ1 + λp)φ

21 + (3λ2 + λp)φ

22

]. (16)

If 3λ1 + λp < 0, the field φ1 develops a nonzero vev vφ1at temperatures above

Ts,crit ≡ Ts(Tcrit) '[12µ21/|3λ1 + λp|

]1/2, (17)

breaking one of the Z2 symmetries. Here, Ts(Tcrit) de-notes the sterile sector temperature at the time when thephoton temperature is Tcrit. The subscript “crit” standsfor critical temperature. One can see that vφ1 6= 0 willoccur for modestly large negative values of the quarticcross-coupling λp. Because of the boundedness condi-tions that force 3λ2 + λp > 0, the other scalar φ2 cannotdevelop a vev simultaneously, so one of the Z2 symme-tries remains unbroken. It is easy to see that if µ21 is verysmall, Tcrit can be quite low, perhaps lower than the tem-

perature Tdec at which active and sterile neutrinos finallydecouple for good.

The charges of sterile neutrinos under the Z2 ×Z2 arechosen as follows: the left-handed component νsL of theDirac fermion νs carries charges (+,+), while its right-handed partner νsR is a singlet with charges (−,+). Thecouplings of νs are then

Ls ⊃ −yφ1νsLνsR −1

2msLν

csLνsL −

1

2msRν

csRνsR + h.c. .

(18)

The two main issues that we discuss now are whetherthis interaction is sufficient to generate a large vφ1-induced thermal mass for νs in order to prohibit νs pro-duction until Tγ < Tdec, and whether sterile neutrinoscattering can freeze-out already at Tγ > Tcrit in orderto avoid recoupling below Tcrit. Of course, we also have todemand that Tcrit < 1 MeV to prevent collisional sterileneutrino production via Z- and W -mediated processes.

At high temperatures, Tγ > Tcrit, the temperature-dependent (Dirac) mass for νs is

ms(Ts) = y

√−12µ21 − (3λ1 + λp)T

2s

12λ1for Tγ > Tcrit .

(19)

At Tγ < Tcrit, νs splits into two Majorana fermions ofmass msL and msR. We assume that at least one ofthese is O(eV). To prohibit production of νs, we demandthat ms(Ts) & Ts at Ts > Ts,crit, so that νs productionbecomes exponentially suppressed.

As an example, if λ2 ' 1 then λ1 & λ2p satisfies the

boundedness criterion, and ms(Ts) ' yTs/√

12|λp| for−1 � λp < 0 and Ts � Ts,crit. The requirement Tcrit <Tdec then implies

|λp| > 12µ21/T

2s,dec , (20)

while the condition ms(Ts) > Ts implies

|λp| <y2

12. (21)

To determine Tdec, we need to consider φ1-mediatedneutrino–neutrino scattering. We first redefine the fieldφ1 → vφ1 + ρ after symmetry breaking, where the massof the physical scalar ρ is given by m2ρ(Ts) = 2λ1v

2φ1 '

T 2s |λp|/6 for Tγ > Tcrit (i.e., Ts > Ts,crit) and m2ρ '

µ21 + (3λ1 + λp)T2s /12 for Tγ ≤ Tcrit. The decoupling

temperature is defined as the temperature at which therate for νa–νs inelastic scattering drops below the Hubblerate:

nν(Ts,dec)sin2 θ0 y

4

m2ρ= H(Tdec) '

T 2decMPl

. (22)

10

��-�� ��-�� ��-�� ��-�� ��-�� ��-����-����-����-����-����-����-��

������� �����-�������� |λ� |

ν ����������������

λ� ≈ λ��� λ� ≈ �� μ�≈ �

μ� [��]��-���-����-��

Figure 5. Parameter regions in which active and sterile neu-trinos never recouple in the toy model given by eqs. (15)and (18). Different colors corresponds to different values ofthe scalar mass parameter µ1, as indicated in the legend. Wehave taken λ1 ' λ

2p, λ2 ' 1, µ2 ' 0 to make this figure.

With nν(Ts) ' T3s and Tdec ≥ Ts,dec, we get

Ts, dec ≥m2ρ

sin2 θ0 y4MPl

. (23)

Therefore, eqs. (20) and (21) are true as long as

12 sin4 θ0 y8 µ

21M

2Pl

m4ρ< |λp| <

y2

12. (24)

This condition can be satisfied if y is tiny, and |λp| iseven tinier.

The parameter space which satisfies all the above con-ditions is shown in fig. 5 for various choices of µ1. We seethat in this minimal toy model the parameters λp andy need to have rather extreme values. Nevertheless, themodel serves as a proof of principle that inverse symme-try breaking provides a viable mechanism for preventingνs production in the early Universe.

To constrain the favored parameter regions of thismodel more quantitatively, it would be necessary tocompute the effective temperature-dependent potentialVeff and follow its evolution through cosmological his-tory, for instance using the public software packageCosmoTransitions [70, 71]. For a given set of modelparameters, this would lead to a prediction for thetemperature-dependent mass of the sterile neutrinos,which could be plugged into the Boltzmann equationsgoverning their production to determine their final abun-dance. The relevant contributions to Veff are, besidesthe tree level terms given in eq. (15), the temperature-independent Coleman-Weinberg corrections [72, 73], theone-loop finite-temperature corrections [74], and the re-sumed higher-order “daisy” terms [75]. As our goal hereis merely to illustrate the phenomenological viability of

sterile neutrino models with inverse symmetry breaking,this computation is far beyond the scope of the presentwork.

This model is an example for a more general classof models exhibiting inverse symmetry breaking, wherethere is greater symmetry at lower temperatures as op-posed to the usual scenario where symmetries are re-stored at higher temperatures [57, 58, 76]. There areother implementations of this mechanism, for instanceWeinberg’s O(N1) × O(N2) scalar models [57], whichbreak to O(N1 − 1) × O(N2) at high temperature. Atthe non-perturbative level the symmetry may get re-stored at very high temperatures and the parameterspace available for such inverse symmetry breaking issmaller than what is suggested by a 1-loop perturbativetreatment [59, 77, 78]. However, for our purposes it issufficient that a phase of broken symmetry exists at in-termediate temperatures.

B. Recoupling happens below MeV but CMBbounds on neutrino mass are avoided

An alternative way of reconciling sterile neutrinos withcosmology is to tolerate their production at Tγ < 1 MeV,but to invoke extra degrees of freedom to evade con-straints on

∑mν .

(i) Extra relativistic degrees of freedom to avoid struc-ture formation bounds. At intermediate couplings(blue/orange vertically striped region in fig. 3), eV-scalesterile neutrinos with secret interactions are constrainedonly by structure formation bounds on

∑mν . One way

to avoid these is to introduce several additional ster-ile states, also charged under U(1)s and with not toosmall mixing with active neutrinos, but with masses� 1 eV. When secret interactions recouple active andsterile states at temperatures < MeV, the energy densityin the neutrino sector is evenly distributed among allneutrino states. If the number of nearly massless statesis sufficiently large, only a small fraction of energy willremain for the eV-scale state. More precisely, by addingn massless states in addition to the three active neutri-nos and the one eV-scale states, the energy density af-ter recoupling will be 3ρSM/(4 + n) in each state, whereρSM is the energy density of each active neutrino fla-vor in the SM. Correspondingly, the effective bound on∑mν is weakened by a factor 3/(4 +n). We see that, in

order to reconcile a 1 eV sterile neutrino with the limit∑mν < 0.23 eV [21], we need to add n ≥ 9 massless

states.(ii) Extra relativistic degree of freedom for enhanced

free-streaming. At large αs (red region in fig. 3, wherethe

∑mν constraint is avoided because νs start to free-

stream only very late, the main problem faced by thevanilla model is that also at least one of the active neutri-nos will start to free-stream too late. Again, this problemcould be avoided by adding one extra relativistic species.This could be for instance a second, nearly massless, ster-

11

ile neutrino that partially thermalizes before neutrino de-coupling, or it could be the A′ boson itself, providedit is nearly massless. This scenario would predict Neffslightly larger than the SM value, but possibly still con-sistent with constraints. On the other hand, the extrafree-streaming provided by the additional species is likelyto improve the fit to CMB and structure formation datanevertheless. A detailed investigation of the viability ofsuch a scenario requires a full fit to CMB and large scalestructure data, which is left for future work.

(iii) Fast sterile neutrino decay. Cosmological con-straints on the secret interactions scenario could alsobe avoided if the eV-scale sterile neutrino decays fastenough. In particular, if νs decays to nearly mass-less states before the onset of structure formation atT ∼ 1 eV, large scale structure observations can hardlyprobe the impact of νs mass. An appealing possibility isto introduce two additional (nearly) massless particles:one pseudo-Goldstone boson φ, and a second sterile neu-trino ν′s. For an interaction vertex of the form yφ(ν̄sγ5ν

′s)

with y & 10−13, the lifetime corresponding to the decayνs → ν

′s + φ is shorter than the time scale of recombina-

tion, avoiding the CMB bounds on both∑mν and Neff.

In scenarios of this type, the strong νs self-interactioninduced by the coupling φ(ν̄sγ5νs) is in itself enough tosuppress νs production before BBN [28]. In other words,φ can take the place of the gauge boson A′ in mediatingsecret interactions. Alternative decay scenarios, such asthree-body decays νs → 2ν

′s+ ν̄

′s (via an off-shell massive

A′) or νs → ν′s + γ cannot generate sufficiently fast νs

decays without violating cosmological constraints [42].

V. SUMMARY

To summarize, we have assessed the status of mod-els featuring light sterile neutrinos νs with “secret” self-interactions mediated by a new gauge boson A′. Suchmodels had originally been introduced as a way of recon-ciling light sterile neutrinos (as motivated by the shortbaseline oscillation anomalies) with cosmological con-straints. Indeed, the effective temperature-dependentpotential generated by secret interactions can efficientlysuppress active–sterile neutrino mixing in the early Uni-verse down to temperatures � MeV. At that time, SMweak interactions have frozen out and the neutrino sectoris fully decoupled from the photon bath, so that the num-ber of relativistic species Neff cannot change any more.However, efficient collisional production of νs (at the ex-pense of active neutrino νa) will occur as soon as themixing angle suppression is lifted. Of particular impor-tance in this context are A′-mediated scattering processeswhich can be strongly enhanced by the s-channel A′ reso-nance (for M ∼ Tν), and by collinear enhancement in theforward direction (for M � Tν). For νs masses around1 eV and vacuum mixing angles of order 0.1 (as mo-tivated by the short-baseline oscillation anomalies), theresulting population of νs is large enough to violate the

cosmological constraint on∑mν . Thus, for ms = 1 eV

and θ0 = 0.1, all values of the A′ mass M and the corre-

sponding fine structure constant αs are disfavored. Ourresults confirm earlier findings from ref. [41]. A possibleloophole to these arguments exists at very large valuesof αs. Namely, if secret interactions are so strong thatνs cannot free-stream, measurements of

∑mν are not

sensitive. However, it has been shown in ref. [42] that inthis case also active neutrino free-streaming is reduced,in conflict with CMB bounds.

In the second part of the paper we have discussed sev-eral new mechanisms for reconciling eV-scale sterile neu-trinos with cosmology. We have outlined a toy model inwhich sterile neutrinos have initially a very large massgenerated by the vacuum expectation value (vev) of anew scalar field, so that their production is kinemati-cally forbidden. Only very late in cosmological history,a phase transition reduces the scalar vev to zero and theνs mass to O(eV). At that time, collisional productionis no longer possible, so the cosmological νs abundanceremains negligible to this day. We have shown that thisscenario is viable, but requires rather extreme values forsome of its coupling constants.

We have in addition discussed scenarios with severalnew relativistic degrees of freedom with masses � eV.As far as cosmological bounds are concerned, these de-grees of freedom behave like active neutrinos. They caneither serve to deplete the νs abundance by equilibratingwith them, or they can act as an extra free-streamingspecies, compensating for the reduced free-streaming ofactive neutrinos in the regime of very large αs.

Finally, we have argued that bounds on∑mν can also

be avoided in scenarios in which the sterile neutrino hasa fast decay mode to active neutrinos plus a light boson.Such models are of particular interest in the context ofthe LSND and MiniBooNE anomalies [17].

ACKNOWLEDGEMENTS

It is a pleasure to thank Alessandro Mirizzi for use-ful discussions. X.C. is supported by the ‘New Frontiers’program of the Austrian Academy of Sciences. BD ispartially supported by the Dept. of Science and Tech-nology of the Govt. of India through a Ramanujam Fel-lowship and by the Max Planck-Gesellschaft through aMax Planck-Partnergroup. JK and NS have been sup-ported by the German Research Foundation (DFG) un-der Grant Grant Nos. KO 4820/11, FOR 2239, EXC-1098 (PRISMA) and by the European Research Coun-cil (ERC) under the European Union’s Horizon 2020 re-search and innovation programme (grant agreement No.637506, “νDirections”). JK would like to thank CERNfor hospitality and support during the final months ofthis project.

12

Table I. Dominant processes and cross sections for production of the mostly sterile mass eigenstate ν4.

process relativistic cross section

(i) e− + e+ → ν̄1 + ν4 σep1̄4 = sin2 θm

πα2

48s4W c

4W

√s√

s−4m2e

(13−20c2W +8c4W )s+(23−40c

2W +16c

4W )m

2e

m4Z

(ii) e− + ν1 → e− + ν4 σe1e4 = sin

2 θmπα

2

24s4W c

4W

(s−m2e)2

m4Z

(31−44c2W +16c4W )s

2−2(7−11c2W +4c4W )sm

2e+4(1−c

2W )

2m

4e

s3

(iii) e+ + ν1 → e+ + ν4 σp1p4 = sin

2 θmπα

2

24s4W c

4W

(s−m2e)2

m4Z

(21−36c4W +16c4W )s

2−(9−18c2W +8c4W )sm

2e+(3−2c

2W )

2m

4e

s3

(iv) ν1 + ν1 → ν1 + ν4 σ1114 = sin2 θm

πα2

s4W c

4W

s

2m4Z

(v) ν̄1 + ν1 → ν̄1 + ν4 σ1114 = sin2 θm

πα2

s4W c

4W

s

3m4Z

(vi) ν̄4 + ν1 → ν̄4 + ν4 σ4144 = sin2 θm

4πα2s

3(3s

2+10M

2s−12M4)s+12M2(s2−M4) log(1+s/M2)M

2s[(s−M2)2+M2Γ2M ]

(vii) ν4 + ν1 → ν4 + ν4 σ4144 = sin2 θm4πα

2s

(s+2M2)s+2M

2(s+M

2) log(1+s/M

2)

M2(s+M

2)(s+2M

2)

Appendix A: Production Rate of Sterile Neutrinos

As discussed in section III, we have taken into accountseven different processes in the computation of the ster-ile neutrino recoupling temperature. Here, in Table I, welist the corresponding cross sections in the 1+1 flavor ap-proximation and to leading order in the mixing angle. Ofcourse, the corresponding CP-conjugate processes con-tribute with the same cross sections. Note that process(iv) has identical initial state particles, a fact that needsto be properly taken into account when computing therate for this process by integrating over the distributionof initial state νe. In process (vi), which can be me-diated by an s-channel A′, we need to take into accountthe non-zero width of A′, which is given by ΓM = αsM/3for Dirac νs.

The production rate of sterile neutrinos, normalized tothe volume element occupied by an active neutrino asexplained in section III A, is given by

Γs =cQZneqνa

∑i

∫d3p1 d

3p2 fi(~p1)fi(~p2)× σi(s)vMøl ,

(A1)

where the sum runs over the seven production processeslisted above, the integral is over the 3-momenta of theinitial state particles, and the prefactor cQZ accounts for

quantum Zeno suppression (see section III A for details).The Møller velocity vMøl reduces to the relative veloc-ity of the two colliding particles in the non-relativisticlimit (see ref. [79] for details). The momentum distribu-tion functions of the initial state particles fi(~p1,2) havea Fermi-Dirac shape as we only consider fermionic pro-cesses. The number density of active neutrinos neqνa is theintegral over the corresponding Fermi-Dirac distributionfor one massless degree of freedom. Abbreviating the in-tegral by introducing the notation 〈·〉 for the momentum-averaged cross section, we obtain eq. (4).

The characteristic temperatures of the initial state par-ticles are Tγ for electrons and positrons, Tν for activeneutrinos, and Ts for sterile neutrinos. As an approxi-mation, we assume the same relation between Tγ and Tνas in the SM. The only exception is that for processes(ii) and (iii), which involve both a charged lepton and aneutrino in the initial state, we set Tν = Tγ despite the

temperature difference between them after e+e− annihi-lation. This approximation, which makes the numericalevaluation of Γs easier, is justified by the fact that afterBBN electrons quickly become decoupled, so that pro-cesses mediated by SM W and Z bosons should not playan important role any longer. In the case of process (iv),which is initiated by two identical particles, care must betaken to restrict the integration domain such that double-counting of initial state phase space is avoided. (Or, al-ternatively, an extra factor of 1/2 needs to be includedin the integrand.)

13

The integrals in eq. (A1) can be partially evaluated [79]. With the definitions E± = E1 ± E2, the result is

Γs =cQZneqνa

2π2∑i

∫ds dE+ dE−fi

(E+ + E−

2

)fi

(E+ − E−

2

)× σi(s)

√[s− (m1 +m2)

2][s− (m1 −m2)2]

s, (A2)

with the integration boundaries

s ≥ (m1 +m2)2 , E+ ≥

√s , (A3)

|E− − E+m21 −m

22

s| ≤

√(E2+ − s)[s− (m1 +m2)

2][s− (m1 −m2)2]

s. (A4)

We use the condition

Γs(Trec) = H(Trec) (A5)

to numerically decide whether and at what recoupling temperature, Trec, the sterile νs can be brought into the thermalequilibrium with active neutrinos.

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Sterile Neutrinos with Secret Interactions — Cosmological Discord?AbstractI IntroductionII The Secret Interactions ScenarioIII Constraints on Sterile Neutrinos with Secret InteractionsA Recoupling Temperature ComputationB Multi-flavor evolutionC Results

IV Reconciling Secret Interactions with CosmologyA Recoupling never happensB Recoupling happens below MeV but CMB bounds on neutrino mass are avoided

V Summary AcknowledgementsA Production Rate of Sterile Neutrinos References