Observing Dirac neutrinos in the cosmic microwave background

Planned CMB Stage IV experiments have the potential to measure the effective number of relativistic degrees of freedom in the early Universe, $N_\text{eff}$, with percent-level accuracy. This probes new thermalized light particles and also constrains possible new-physics interactions of Dirac neutrinos. Many Dirac-neutrino models that aim to address the Dirac stability, the smallness of neutrino masses or the matter--anti-matter asymmetry of our Universe endow the right-handed chirality partners $\nu_R$ with additional interactions that can thermalize them. Unless the reheating temperature of our Universe was low, this leads to testable deviations in $N_\text{eff}$. We discuss well-motivated models for $\nu_R$ interactions such as gauged $U(1)_{B-L}$ and the neutrinophilic two-Higgs-doublet model, and compare the sensitivity of SPT-3G, Simons Observatory, and CMB-S4 to other experiments, in particular the LHC.


I. INTRODUCTION
The sensitivity of anisotropies in the cosmic microwave background (CMB) to extra radiation density like that in the form of effective extra numbers of neutrinos N eff has been known for some time [1]. Upcoming limits from the CMB and large-scale structure on extra radiation from the early Universe are entering a qualitatively new regime, with sensitivity to particle species that have decoupled from equilibrium at very early times and high energy scales. In this article, we show that there are direct implications of this sensitivity to neutrino mass models.
Any extra radiation energy density is usually normalized to the number density of one active neutrino flavor, N eff ≡ (8/7) (11/4) 4/3 ρ ν /ρ γ . The current Planck measurement is N eff = 2.99 ± 0.17 (including baryon acoustic oscillation (BAO) data) [2], perfectly consistent with the Standard Model (SM) expectation N SM eff = 3.045 [3][4][5]. CMB Stage IV (CMB-S4) experiments have the potential to constrain ∆N eff ≡ N eff − N SM eff = 0.060 (at 95% C.L.) [6,7], which is very sensitive to new light degrees of freedom that were in equilibrium with the SM at some point, even if it decoupled at multi-TeV temperatures. Indeed, a relativistic particle φ that decouples from the SM plasma at temperature T dec contributes ∆N eff 0.027 106.75 g (T dec ) where g s is the number of spin degrees of freedom of φ (multiplied by 7/8 for fermions) and g (T dec ) is the sum of all relativistic degrees of freedom at T = T dec . At temperatures above the electroweak scale, g saturates to 106.75, the maximum amount of entropy available from SM particles. Ref. [8] has recently studied the impact of CMB-S4 on axions and axion-like particles (g s = 1), which are reasonably well motivated but could easily lead to an entropy-suppressed contribution ∆N eff 0.027 that is below the CMB-S4 reach. It should be kept in mind, however, that an even better motivation for light degrees of freedom comes from the discovery of non-zero neutrino masses: if neutrinos * kevork@uci.edu † Julian.Heeck@uci.edu are Dirac particles then we necessarily need two or three effectively massless chirality partners ν R in our world, which would contribute a whooping ∆N eff ≥ 2 × 0.047 = 0.09 (two ν R ) or even ∆N eff ≥ 0.14 (three ν R ) if thermalized with the SM, easily falsifiable or detectable! While it is well known that just SM + Dirac ν does not put ν R in equilibrium due to the tiny Yukawa couplings m ν / H 10 −11 [9,10], one often expects additional interactions for ν R in order to explain the smallness of neutrino masses, to generate the observed matter-antimatter asymmetry of our Universe, and to protect the Dirac nature from quantum gravity. All of these new ν R interactions will then face strong constraints from CMB-S4 that will make it difficult to see the mediator particles in any other experiment, in particular at the LHC.
The basic idea to measure new interactions via N eff in Big Bang nucleosynthesis (BBN) or the CMB is old [11,12], of course, see for example the reviews [13][14][15]. It is timely to revisit these limits though since we are on the verge of reaching an important milestone: sensitivity to Dirac-neutrino induced ∆N eff even if the ν R decoupled above the electroweak phase transition! As we will outline in this article, the non-observation of any ∆N eff in CMB-S4 will then have serious consequences for almost all Dirac-neutrino models, in particular those addressing the origin of the small neutrino mass.
The rest of this article is organized as follows: Sec. II gives a brief overview of the current measurements of N eff and future reach. In Sec. III we discuss the impact of stronger ∆N eff limits on a number of Dirac-neutrino mass models. We conclude in Sec. IV.

II. OBSERVING N eff
The CMB is sensitive to the radiation energy density of the Universe via the variant effects of radiation on the features of the acoustic peaks of the CMB and its damping tail. The acoustic scale of the CMB is altered inversely proportionally to the Hubble rate at the time of last scattering, θ sound ∝ H −1 , while the scattering causing the exponentially-suppressed damping tail of the CMB anisotropies goes as θ damping ∝ H −1/2 . These differential effects provide the primary signatures of extra ∆N eff in the CMB power spectrum. The primordial helium abundance, Y p , also changes the scales of θ sound to θ damping similarly, however the near degeneracy between N eff and Y p is broken by other physical effects, including the early integrated Sachs-Wolfe effect, effects of a high baryon fraction, as well as the acoustic phase shift of the acoustic oscillations [16,17].
The limit from Planck plus BAO data is N eff = 2.99 ± 0.17 [2], where the limit is from a single parameter extension of the standard ΛCDM 6-parameter cosmological model. We translate this into a 2σ constraint ∆N eff < 0.28. Currently underway and future experiments are forecast to have even greater sensitivity, even with more conservative assumptions about the possible presence of new physics. The South Pole Telescope SPT-3G is a ground-based telescope currently in operation, with a factor of ∼ 20 improvement over its predecessor. SPT-3G is forecast to have a sensitivity of σ(∆N eff ) = 0.058, given here as the single standard deviation (1σ) sensitivity [18]. This sensitivity is conservative in that it includes the variation of a nine-parameter model for all of the new physics which SPT-3G will be tackling: ΛCDM (six parameters), N eff , active neutrino mass (Σm ν ), plus tensors. We estimate the 2σ sensitivity of SPT-3G as ∆N eff < 0.12. The CMB Simons Observatory (SO), which will see first light in 2021, is forecast to have 1σ sensitivity in the range of σ(∆N eff ) = 0.05 to 0.07 [19].
For the noise level and resolution of CMB-S4, the differential effects on the acoustic peaks and damping tail are predominately measured through the T E spectrum at multipoles > 2500 [7]. The sensitivity of CMB-S4 is forecast to be ∆N eff = 0.060 at 95% C.L., as a single parameter extension to ΛCDM.
In Fig. 1 we show the current 2σ limit on N eff as well as the SPT-3G, SO, and CMB-S4 forecast as a function of the decoupling temperature T dec using Eq. (1). The current Planck limit requires T dec 0.55 GeV for three right-handed neutrinos, whereas SPT-3G, SO, and CMB-S4 can conclusively probe this scenario for arbitrary decoupling temperatures! If only two ν R are in equilibrium, then SPT-3G/SO can probe T dec ∼ 30 GeV and CMB-S4 is required to reach arbitrary decoupling tem-peratures. It is then clear that SPT-3G, SO, and CMB-S4 provide a significant sensitivity to the new physics of Dirac-neutrino models.

III. IMPACT ON DIRAC NEUTRINO MODELS
In the following we will discuss the impact a nearfuture constraint ∆N eff < 0.06 would have on models involving Dirac neutrinos, which automatically bring two to three relativistic states ν R that could be in equilibrium and contribute to N eff . As with all constraints from cosmology, our conclusions rest on additional assumptions regarding the cosmological evolution, namely 1. We assume general relativity and the cosmological standard model ΛCDM.
2. We assume that the (reheating) temperature of the Universe reached at least the mass of the particles that couple to ν R . This is a strong assumption since we technically only know that the Universe was at least ∼ 5 MeV hot [20], everything beyond being speculation. Note however that most solutions to the matter-anti-matter asymmetry require at least electroweak temperatures in order to thermalize sphalerons. Dark matter production also typically requires TeV-scale temperatures, at least for weakly interacting massive particles.
3. No significant entropy dilution. To dilute three ν R down to ∆N eff < 0.06 via Eq. (1) one would need to roughly double the SM particle content. This means that Dirac neutrinos would evade N eff constraints if they decoupled at temperatures above the hypothetical supersymmetry or grand-unifiedtheory breaking scales, as both of these SM extensions bring a large number of new particles with them. A different way to generate entropy comes from an early phase of matter domination, which requires a heavy particle that goes out of equilibrium while relativistic and then decays sufficiently late so it has time to dominate the energy density of the Universe [21,22].
Note that even if the ν R never reached equilibrium, it is possible that they were created non-thermally and still leave an imprint in N eff [23]. Following Refs. [23][24][25] it might even be possible to distinguish this ν R origin of ∆N eff by observation of the cosmic neutrino background, e.g. with PTOLEMY [26]. This will not be discussed here.
On a final note, we will restrict our discussion to renormalizable UV-complete quantum field theories. An alternative approach would be to study higher-dimensional operators of an effective field theory with SM fields + Dirac-ν and put constraints on the Wilson coefficients, e.g. on the Dirac-ν magnetic moments [27][28][29][30][31]. However, higher-dimensional operators will give ν R production rates that are dominated by the highest available temperature and thus depend explicitly on it [8]. In any renormalizable realization of such operators this growing rate would be cured once the underlying mediators to into equilibrium, which then brings us back to the approach pursued here. One important task of Dirac-neutrino model building is to protect the Dirac nature, i.e. to forbid any and all ∆L = 2 Majorana mass terms for the neutrinos. While this can easily be achieved by imposing a global lepton number symmetry U (1) L on the Lagrangian, there is the looming danger that quantum gravity might break such global symmetries [32]. To protect the Dirac nature from quantum gravity it might then be preferable to use a gauge symmetry to distinguish neutrino from anti-neutrino. The simplest choice is U (1) B−L , which is already anomaly-free upon introduction of the three ν R that we need for Dirac neutrino masses. For unbroken U (1) B−L the Z gauge boson can still have a Stückelberg mass, a scenario discussed in Refs. [33,34]. 1 In a more extended scenario one can even break U (1) B−L spontaneously, as long as it is by more than two units in order to forbid Majorana mass terms [39]. The simplest example given in Ref. [40] has a spontaneous symmetry breaking U (1) B−L → Z 4 , where the remaining discrete gauge symmetry protects the Dirac nature of the neutrinos and the ∆(B − L) = 4 interactions allow for leptogenesis [41], as discussed below. This broken U (1) B−L scenario also allows an embedding into larger gauge groups such as left-right, Pati-Salam or SO(10) [39].
is attained when this rate Γ exceeds the Hubble rate H(T ) ∼ T 2 /M Pl at a certain temperature. The behavior of Γ/H(T ) is shown in Fig. 2, using the formulae from Ref. [34]. As can be seen, the ratio Γ/H(T ) is largest at the temperature T ∼ M Z /3, where inverse decays of Z are highly efficient, so the most aggressive assumption is that the Universe reached this temperature. Notice that a light Z will itself start to contribute to N eff [46,47]. For heavy Z masses above 20 GeV, we demand that the ν R go out of equilibrium before T ∼ 0.5 GeV (Fig. 1), which corresponds to the constraint M Z /g > 14 TeV, far better than pre-Planck limits [34,[42][43][44][45]. A similar limit was recently derived in Ref. [48]. For masses MeV < M Z 10 GeV the limit becomes much stronger due to the s-channel resonance of the rate, or equivalently the efficient inverse decay of Z . Here we demand that the ν R are out of equilibrium for all temperatures between MeV and T ∼ 0.5 GeV. For Z masses below MeV it becomes possible for the ν R to go into equilibrium below T ∼ MeV, leaving BBN unaffected. However, even in this case the thermalization of ν L , ν R , and Z after ν L decoupling would leave an impact on N eff [49,50], already excluded by CMB data. As a result, we have to forbid ν R /Z thermalization for all temperatures between eV (CMB formation) and T ∼ 0.5 GeV, which gives the black exclusion line in Fig. 3, updating Ref. [34].
This existing N eff constraint is stronger than most laboratory experiments, except for dilepton searches at the LHC. If future measurements in SPT-3G, SO, and CMB-S4 push the ∆N eff bound below 0.14, the limits on Z will change dramatically to shown as a red dashed line in Fig. 3, because we have to demand that the ν R were never in equilibrium with the SM. Once again, this limit assumes that the Universe reached a temperature of at least T ∼ M Z /3, otherwise the bound weakens. Keeping these assumptions in mind it is clear from Fig. 3 that the non-observation of ∆N eff in future CMB experiments will make it impossible to find a Z coupled to Dirac neutrinos in any laboratory experiment. Turning this around, the observation of a U (1) B−L gauge boson in a collider or scattering experiment would then prove that neutrinos are Majorana particles. This conclusion is not limited to B − L but extends to other Z [42-45, 48, 62] or W [11,12,63] models. In general, new gauge interactions of ν R will face strong constraints from CMB-S4 that will make it difficult to see the gauge bosons, say Z or W R , in any other experiment, in particular at the LHC.

B. Neutrinophilic 2HDM and other mass models
Extending the SM by two or three gauge singlets ν R allows for Yukawa couplings with the Brout-Englert-Higgs doublet H which give a Dirac-neutrino mass matrix m ν = y H after electroweak symmetry breaking. The overall neutrino mass scale is still unknown, but upper bounds between 0.1 eV and 0.2 eV can be obtained from cosmology [64], which in turn require us to consider Yukawa couplings y = m ν / H 10 −12 . This is a million times smaller than the already-small electron Yukawa and is considered an unappealing finetuning by most theorists [65]. This has spawned a vast literature of models that generate small Dirac masses via other mechanisms and most importantly without the use of small couplings. The general idea is to forbid the coupling of Eq. (3) by means of an additional symmetry [62,65,66] and instead couple ν R to new mediator particles that eventually also couple to ν L and thus create a Dirac mass, often suppressed by loop factors or mediator mass ratios.
The crucial point is that the new mediator particles unavoidably couple to ν L with non-tiny couplings, which thermalizes them in the early Universe at temperatures around their mass. In order to connect ν L to ν R , some of the mediators also have gauge interactions under SU (2) L × U (1) Y . Since they also have non-tiny couplings with ν R by construction, this puts the ν R in thermal equilibrium with the SM. Unlike the Z model of the previous section it makes little sense here to consider couplings that are too small to reach ν R equilibrium, as this would defeat the purpose of these models. The only way to evade N eff constraints is then to assume that the Universe never reached temperatures of order of the mediator masses.
As an explicit and rather minimal example let us consider the neutrinophilic two-Higgs-doublet model (2HDM) [67][68][69][70][71], which introduces a second scalar doublet φ that exclusively couples to ν R by means of a new symmetry: All charged fermions obtain their mass from the main doublet H with vacuum expectation value around 174 GeV, but the neutrinos obtain a Dirac mass m ν = κ φ . Instead of using small Yukawa couplings it is then possible to simply have a smaller vacuum expectation value for the neutrinos, e.g. φ ∼ eV. The Yukawa couplings κ can then be large, apparently resolving the unwelcome finetuning of Eq. (3). Of course, the κ will in particular be large enough to thermalize the ν R , seeing as φ is an electroweak doublet that is certainly in equilibrium with the SM at temperatures around m φ [71]. The neutrinophilic 2HDM thus predicts ∆N eff > 0.09 (two ν R ) or ∆N eff > 0.14 (three ν R ) unless the temperature never reached m φ . In general, any renormalizable model that aims to explain why the Dirac neutrino masses are so small does so by introducing new mediator particles. The couplings of these mediators to ν R and ν L are not tiny by construction, so will thermalize the ν R if the temperature ever reached the mass of the mediators. Generically we then expect a contribution to ∆N eff in any model that addresses m ν m W .

C. Leptogenesis
Above we have argued that Dirac neutrinos could have additional interactions based on rather theoretical motivations such as Dirac stability and the smallness of neutrino masses. There is however a more pressing issue that any model of Dirac neutrinos needs to address: the baryon asymmetry of our Universe. For Majorana neutrinos there exist a variety of leptogenesis scenarios, in which CP-violating, out-of-equilibrium processes with ∆L = 2 generate a lepton asymmetry that is then transferred to a baryon asymmetry via ∆(B+L) = 6 sphaleron processes. For Dirac neutrinos, there exist essentially two variations of leptogenesis: • Neutrinogenesis [72,73]: without ever breaking B− L, we let a new particle σ decay out-of-equilibrium into ν R and left-handed leptons in such a way that a lepton asymmetry is generated in the ν R that is exactly opposite to an asymmetry in the lefthanded leptons: ∆ ν R = −∆ L = 0. If the ν R are not thermalized afterwards a baryon asymmetry is generated out of ∆ L by the sphalerons.
• Dirac leptogenesis [41]: breaking B − L by any unit n other than two makes it possible to protect the Dirac nature of neutrinos but still create a lepton asymmetry via ∆(B − L) = n interactions in complete analogy to Majorana leptogenesis mechanisms. In the simplest example with n = 4 one creates a lepton asymmetry in ν R via CP-violating, out-of-equilibrium decays of a new particle ψ → ν R ν R ,ν RνR . This asymmetry in ν R now needs to be transferred to the left-handed leptons, e.g. via new Yukawa interactionsLφν R , in order to be further processed by the sphalerons.
From the above it is clear that Dirac leptogenesis [41] in its simplest form strongly requires thermalized ν R , e.g. via a neutrinophilic 2HDM, and thus predicts ∆N eff ≥ 0.14. Neutrinogenesis [72] on the other hand requires the ν R to be out-of-equilibrium after the asymmetry generation, but unavoidably has them thermalized before, when the mother particle σ was still in equilibrium. 2 Here, too, we thus expect a contribution to N eff . In general we thus expect a ν R contribution to N eff from any leptogenesis mechanism with Dirac neutrinos. The usual mechanisms used to evade this contribution -additional entropy dilution or a temperature below the mediator particle -would also render leptogenesis more inefficient. Therefore, if CMB-S4 does not observe a ∆N eff it is probably necessary to consider baryogenesis mechanisms that do not involve leptons.

IV. CONCLUSION
Measurements of the radiation density in the early Universe, usually parametrized via the effective number of neutrino species N eff = 3.045+∆N eff , have reached an astonishing precision within the last decade or so, thanks to experiments such as Planck. The ongoing SPT-3G experiment and the future Simons Observatory and CMB-S4 experiment will further increase our knowledge and reach sensitivities down to ∆N eff 0.06 (95% C.L.). This makes it possible to detect or exclude new ultralight particles even if they decoupled very early in the Universe.
Here we argued that one of the best motivations for such light particles comes from the observation of neutrino oscillations. Indeed, if neutrinos are Dirac particles just like all other known fermions, we have to extend the Standard Model by two or three practically massless chirality partners ν R . Models that aim to address the Dirac stability, the smallness of neutrino masses, or the matter-antimatter asymmetry of our Universe typically endow the ν R with additional interactions that could lead to a thermalization in the early Universe and hence a measurable contribution to N eff . The non-observation of any ∆N eff in upcoming experiments will therefore place strong constraints on Dirac-neutrino models, as illustrated here in some concrete examples. On a more optimistic note, it is entirely possible that Dirac neutrinos will make themselves known in CMB N eff measurements long before their nature is confirmed in more direct ways.