Consider any theory where neutrinos acquire Majorana masses at or close to the weak scale. Since Majorana neutrino masses violate lepton number this implies the existence of new sources of lepton number violation at these energies. What is the effect of this on any pre-existing baryon asymmetry? The interactions that give rise to neutrino masses tend to erase any asymmetry in L. At the same time sphalerons erase any asymmetry in B + L. The combination of these two processes can potentially erase any pre-existing baryon asymmetry. Requiring that the baryon asymmetry survive leads to constraints on the allowed parameter space of these theories, with implications for the LHC. Alternatively, these constraints can be evaded if the pre-existing baryon asymmetry is very large, > 10 -3, or the baryon asymmetry is generated at or below the weak scale Either of these possibilities is interesting in and of itself.
Specifically, the theories we will be focusing on are Type I seesaw models with weak scale right-handed neutrinos and a weak scale Z’ under which the neutrinos are charged Type II seesaw models with a single weak scale SU(2) Higgs triplet Type III seesaw models with fermionic SU(2) triplets Although these arguments are very general and apply to a large class of models, for concreteness we limit ourselves here to seesaw models. Although very similar bounds apply to Type I models without a Z’, we focus on the case with a Z’ for two reasons. A weak scale Z’ is itself one of the best motivated scenarios for beyond the Standard Model physics, and arises in many different contexts. In such a scenario the simplest mechanism to generate neutrino masses is a Type I seesaw, with some of the states necessary to cancel the U(1) 3 anomalies playing the role of right-handed neutrinos. This case is far more interesting for the LHC.
As a concrete example of how the baryon asymmetry is erased consider a toy Type I model with a single lepton flavor. The asymmetry is erased by lepton number violating decays and inverse decays of the right-handed Majorana neutrino. For M of order TeV, the Yukawa coupling λ ~ 10 -6.
At temperatures T > M, the rate for lepton number violating decays and inverse decays of N is given by These processes are in equilibrium if this is larger than expansion rate. For m ν of order the atmospheric scale of 0.05 eV, Γ > H for T ~ M. The baryon asymmetry in this toy model is washed out unless M is below the scale at which the sphalerons freeze out ~ 150 GeV.
However, whether the baryon asymmetry is erased in a realistic theory is a far more detailed question. There are several reasons. The medium distinguishes between L e, L μ and L τ at the weak scale. Since sphalerons conserve for each lepton flavor α separately, lepton flavor violating processes must be in equilibrium for each lepton flavor in order for the baryon asymmetry to be sufficiently erased. The full spectrum of neutrino masses and mixings must be taken into account!
The extent to which the baryon asymmetry is erased depends on the actual period of time that the lepton flavor violating processes are in equilibrium. For m ν of order the atmospheric scale these processes are in equilibrium only for a short time. It then requires a careful analysis to determine how much of the original asymmetry survives. The relevant Boltzmann equation has the general form Here Y B/3 – Lα is the (B/3 – L α ) number density over entropy density, z = M/T and W α is a washout term that is proportional to the rate of decays and inverse decays, and inversely proportional to the Hubble rate. The solution to this equation is If integral is large we get exponential washout!
The baryon asymmetry, having been erased, can be regenerated by out-of-equilibrium decays of the right-handed neutrinos. If the masses of the right-handed neutrinos are of order the weak scale, the Yukawa couplings are generally too small to allow a sizeable baryon asymmetry to be generated this way. However, an exception to this rule occurs if the right-handed neutrinos are extremely degenerate, at the level of one part in 10 10. In this case resonant leptogenesis can occur. In what follows I will assume that the right-handed neutrinos are not sufficiently degenerate to allow this possibility, leaving this special case for further study.
The equation which governs washout for the lepton flavor α is There are three right-handed neutrinos N i with masses M i where i = 1, 2, 3 and the washout parameter receives contributions from each. Each of the W iα is in turn proportional to the rate at which N i decays into the corresponding flavor α and inversely proportional to the Hubble expansion rate, H. Here m * = 1.08 X 10 -3 eV and v is the electroweak VEV.
The precise dependence of W on K is shown in the figure below.
The next step is to correlate the K iα to the neutrino masses. Use the Casas-Ibarra parametrization Here U is the PMNS matrix, D m and D M are diagonal matrices for the light and heavy neutrinos respectively, and Ώ is a complex othogonal matrix. Using this parametrization Numerically scan over Ώ, and over the unknown phases in the PMNS matrix, for the cases of quasi-degenerate, hierarchical and inverse hierarchical neutrino spectra and determine the extent of washout in each case.
In the case of a quasi-degenerate spectrum the washout is large. Specifically the washout is larger than 10 6 in 99% of parameter space unless at least one of the right-handed neutrinos is lighter than about 300 GeV. This case is very promising for the LHC! If the spectrum exhibits inverted hierarchy, the washout is larger than 10 6 in 99% of parameter space unless at least one of the right-handed neutrinos is lighter than 1 TeV. Also promising. If the spectrum exhibits normal hierarchy, the washout is only of order 10-100 in the e flavor in most of parameter space. In this case there is no upper bound on the mass of the lightest right-handed neutrino. However, in the case of the normal hierarchy, at least one of the three right-handed neutrinos is electrophobic, decaying predominantly into muons and taus, rather than into electrons. This merely reflects the fact that the washout in the e flavor is small.
In the best case scenario the Z’ can be produced on-shell, and in turn decays into on-shell right-handed neutrinos. Each right-handed neutrino (dominantly) decays into lepton + W. Since the N’s are Majorana, the leptons can be same sign! The striking signature is like-sign leptons with no missing energy, together with invariant mass peaks. No genuine SM background. What do these results mean for the LHC? The LHC could uncover the mechanism of neutrino mass generation!
However, it is crucial for discovery that the condition M Z ’ > 2 M be satisfied. Otherwise the decay Z’ 2 N is kinematically forbidden, and must proceed off-shell, severely limiting the reach. The mass bounds imply that this condition is very likely to be satisfied if the spectrum is quasi-degenerate or exhibits an inverted hierarchy.
The Type III seesaw has the same parameters as the Type I seesaw. The crucial difference is that instead of SU(2) singlet right-handed neutrinos, the mediators are instead SU(2) triplet fermions Δ. The washout parameter W is again proportional to the ratio of the decay rate to the Hubble rate. The crucial difference is that it is a factor of 3 larger than in the Type I case, because inverse decays and decays of the charged states in Δ also contribute to washout. The constraints are correspondingly tighter. For a quasi-degenerate spectrum of neutrinos at least one triplet must be lighter than 170 GeV, while for an inverted hierarchy the bound is 300 GeV. The reach of the LHC for SU(2) triplets is 1 TeV very promising! For the case of normal hierarchy, there are again no constraints on the triplet mass, but at least one triplet must decay electrophobically.
In a Type II seesaw the neutrinos get their mass from the VEV of the neutral component of a scalar SU(2) triplet Δ. Here α, β run over e, μ and τ. The neutrino mass is given by Here M Δ is the mass of the scalar triplet. A single triplet is enough to generate all the neutrino masses, in contrast to Type I and Type III. We focus on the case that the mass of Δ is in the 100 GeV – TeV range, and therefore accessible to the LHC. Depending on the relative sizes of μ and h, Δ will either decay into Higgs pairs or into leptons. Since the neutrino masses depend only on a combination of these parameters, we scan over μ ≡ v Δ.
How is the baryon asymmetry erased in this model? Note that lepton number is violated only in the presence of both interaction terms below. The washout dominantly arises from LL HH through on-shell Δ. When we require that the washout not exceed 10 6, a substantial portion of the parameter space is disfavored.
In theories where the Standard Model neutrinos acquire Majorana masses at the weak scale, requiring that any pre-existing baryon asymmetry not be washed out can be used to constrain the allowed parameter space. In the case of Type I seesaw models with a weak scale Z’, if the neutrino masses are quasi-degenerate or exhibit an inverted hierarchy, upper bounds can be placed on the mass of at least one right-handed neutrino. This is promising for the LHC. If the masses exhibit normal hierarchy, at least one of the right-handed neutrinos must decay electrophobically. Similar constraints hold for Type III, except the bounds are tighter. In the case of Type II, substantial parameter space is disfavored. These constraints can be evaded if the pre-existing asymmetry is large, of order 10 -3 or more, if the baryon asymmetry is generated at low scales, or if resonant leptogenesis occurs.