Neutrino Mass Seesaw at
the Weak Scale, the
Baryon Asymmetry, and the LHC
Z. Chacko
University of Maryland, College Park
S. Blanchet, R.N. Mohapatra
Introduction
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.
Although these arguments are very general and apply to a large class
of models, for concreteness we limit ourselves here to seesaw models.
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 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.
For M of order TeV, the Yukawa coupling λ ~ 10-6.
The asymmetry is erased by lepton number violating decays and
inverse decays of the right-handed Majorana neutrino.
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 mν 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 Le, Lμ and Lτ 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 mν 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 – Lα) number density over entropy
density, z = M/T and Wα 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 1010.
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.
Type I Seesaw
The equation which governs washout for the lepton flavor α is
There are three right-handed neutrinos Ni with masses Mi where i =
1, 2, 3 and the washout parameter receives contributions from each.
Each of the Wiα is in turn proportional to the rate at which Ni 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 Kiα to the neutrino masses. Use the
Casas-Ibarra parametrization
Here U is the PMNS matrix, Dm and DM 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 106 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 106 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.
What do these results mean for the LHC?
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.
The LHC could uncover the mechanism of neutrino mass generation!
However, it is crucial for discovery that the condition MZ’ > 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.
Type III Seesaw
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.
Type II Seesaw
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 MΔ 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 ≡ vΔ.
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 106, a substantial
portion of the parameter space is disfavored.
Conclusions
• 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.