Broadening the Higgs Boson with Right-Handed Neutrinos and a Higher
  Dimension Operator at the Electroweak Scale
RUNHETC-04-2007
Broadening the Higgs Boson with Right-Handed Neutrinos
and a Higher Dimension Operator at the Electroweak Scale
Michael L. Graesser
1Department of Physics and NHETC, Rutgers University, Piscataway, NJ 08540
Abstract
The existence of certain TeV suppressed higher-dimension operators may open up
new decay channels for the Higgs boson to decay into lighter right-handed neutrinos.
These channels may dominate over all other channels if the Higgs boson is light. For a
Higgs boson mass larger than 2mW the new decays are subdominant yet still of interest.
The right-handed neutrinos have macroscopic decay lengths and decay mostly into final
states containing leptons and quarks. A distinguishing collider signature of this scenario
is a pair of displaced vertices violating lepton number. A general operator analysis is
performed using the minimal flavor violation hypothesis to illustrate that these novel
decay processes can occur while remaining consistent with experimental constraints
on lepton number violating processes. In this context the question of whether these
new decay modes dominate is found to depend crucially on the approximate flavor
symmetries of the right-handed neutrinos.
http://arxiv.org/abs/0704.0438v2
1 Motivation
Neutrino interactions with other Standard Model particles are well-described, forming a
cornerstone of the Standard Model itself. But the origin of their masses remain unknown.
If their masses are generated by a local quantum field theory, then other degrees of freedom
must exist. These particles or “right-handed neutrinos” are either the missing Dirac part-
ners of the neutrinos, or are much heavier than the O(eV) mass scale of the neutrinos and
create a “see-saw” mechanism. Which of these scenarios is realized in Nature is dependent
on the unknown scale of the Majorana mass parameters of the right-handed neutrinos.
The existence of right-handed neutrinos may have other physical consequences, de-
pending on the size of their Majorana masses. Right-handed neutrinos with Majorana
masses violate overall lepton number, which may have consequences for the origin of the
observed baryon asymmetry. Leptogenesis can occur from the out of equilibrium decay of a
right-handed neutrino with mass larger than the TeV scale [1]. Interestingly, right-handed
neutrinos with masses below the electroweak scale may also lead to baryogenesis [2].
But if right-handed neutrinos exist, where did their mass come from? The Majorana
mass parameters are not protected by the gauge invariance of the Standard Model, so an
understanding of the origin of their mass scale requires additional physics. The see-saw
mechanism with order unity Yukawa couplings prefers a large scale, of order 1013−14 GeV.
But in this case a new, intermediate scale must be postulated in addition to the four mass
scales already observed in Nature. On the other hand, such a large scale might occur
naturally within the context of a Grand Unified Theory.
Here I explore the consequences of assuming that the Majorana neutrino mass scale is
generated at the electroweak scale 1. To then obtain the correct mass scale for the left-
handed neutrinos from the “see-saw” mechanism, the neutrino Yukawa couplings must be
tiny, but not unreasonably small, since they would be comparable to the electron Yukawa
coupling. It might be natural for Majorana masses much lighter than the Planck or Grand
Unified scales to occur in specific Randall-Sundrum type models [7] or their CFT dual
descriptions by the AdS/CFT correpsondance [8]. But as the intent of this paper is to
be as model-independent as possible, I will instead assume that it is possible to engineer
electroweak scale Majorana masses and use effective field theory to describe the low-energy
theory of the Higgs boson and the right-handed and left-handed (electroweak) neutrinos.
I will return to question of model-building in the concluding section and provide a few
additional comments.
With the assumption of a common dynamics generating both the Higgs and right-
handed neutrino mass scales, one may then expect strong interactions between these par-
ticles, in the form of higher dimension operators. However since generic flavor-violating
higher dimension operators involving Standard Model fields and suppressed only by the
TeV are excluded, I will use throughout the minimal flavor violation hypothesis [9, 10, 11] in
1For previous work on the phenomenology of electroweak scale right-handed neutrinos, see [3, 4, 5, 6].
None of these authors consider the effects of TeV-scale suppressed higher dimension operators.
order to suppress these operators. The purpose of this paper is to show that the existence
of operators involving the Higgs boson and the right-handed neutrinos can significantly
modify the phenomenology of the Higgs boson by opening a new channel for it to decay
into right-handed neutrinos. I show that the right-handed neutrinos are long-lived and
generically have macroscopic decay lengths. For reasonable values of parameters their de-
cay lengths are anywhere from fractions of a millimeter to tens of metres or longer if one
of the left-handed neutrinos is extremely light or massless. As they decay predominantly
into a quark pair and a charged lepton, a signature for this scenario at a collider would
be the observation of two highly displaced vertices, each producing particles of this type.
Further, by studying these decays all the CP -preserving parameters of the right-handed
and left-handed neutrinos interactions could be measured, at least in principle.
A number of scenarios for new physics at the electroweak scale predict long-lived parti-
cles with striking collider features. Displaced vertices due to long-lived neutral particles or
kinks appearing in charged tracks are predicted to occur in models of low energy gauge me-
diation [12]. More recently models with a hidden sector super-Yang Mills coupled weakly
through a Z ′ or by mass mixing with the Higgs boson can produce dramatic signatures with
displaced jets or leptons and events with high multiplicity [13]. A distinguishing feature of
the Higgs boson decay described here is the presence of two displaced vertices where the
particles produced at each secondary vertex violate overall lepton number.
That new light states or operators at the electroweak scale can drastically modify Higgs
boson physics has also been recently emphasized. Larger neutrino couplings occur in a
model with nearly degenerate right-handed neutrino masses and vanishing tree-level active
neutrino masses, that are then generated radiatively at one-loop [3]. Decays of the Higgs
boson into a right-handed and left-handed neutrino may then dominate over decays to
bottom quarks if the right-handed neutrinos are heavy enough. Models of supersymmetry
having pseudoscalars lighter than the neutral Higgs scalar may have exotic decay processes
for the Higgs boson that can significantly affect limits and searches [14]. Supersymmetry
without R-parity can have striking new signatures of the Higgs boson [15]. Two common
features between that reference and the work presented here is that the Higgs boson decays
into a 6-body final state and may be discovered through displaced vertices, although the
signatures differ.
Interesting phenomena can also occur without supersymmetry. Adding to the Standard
Model higher dimension operators involving only Standard Model fields can modify the
Higgs boson production cross-section and branching fractions [16]. Such an effect can
occur in models with additional colored scalars coupled to top quarks [17].
The outline of the paper is the following. Section 2 discusses the new decay of the Higgs
boson into right-handed neutrinos. Section 3 then discusses various naturalness issues
that arise in connection with the relevant higher dimension operator. Section 4 discusses
predictions for the coefficients of the new operator within the framework of minimal flavor
violation [9, 10, 11]. It is found that the predicted size of the higher dimension operators
depends crucially on the approximate flavor symmetries of the right-handed neutrinos. How
this affects the branching ratio for the Higgs boson to decay into right-handed neutrinos
is then discussed. Section 5 computes the lifetime of the right-handed neutrinos assuming
minimal flavor violation and discusses its dependence on neutrino mass parameters and
mixing angles. I conclude in Section 6 with some comments on model-building issues and
summarize results.
2 Higgs Boson Decay
The renormalizable Lagrangian describing interactions between the Higgs doubletH (1,2)−1/2,
the lepton SU(2)W doublets Li (1,2)−1/2, and three right-handed neutrinos NI (1,1)0 is
given by
mRNN + λνH̃NL+ λlHLe
c (1)
where flavor indices have been suppressed and H̃ ≡ iτ2H∗ where H has a vacuum ex-
pectation value (vev) 〈H〉 = v/
2 and v ≃ 247 GeV. Two-component notation is used
throughout this note. We can choose a basis where the Majorana mass matrix mR is di-
agonal and real with elements MI . In general they will be non-universal. It will also be
convenient to define the 3× 3 Dirac neutrino mass mD ≡ λνv/
2. The standard see-saw
mechanism introduces mass mixing between the right-handed and left-handed neutrinos
which leads to the active neutrino mass matrix
λTν m
R λνv
2 = mTDm
R mD . (2)
This is diagonalized by the PMNS matrix UPMNS [18] to obtain the physical masses mI
of the active neutrinos. At leading order in the Dirac masses the mass mixing between the
left-handed neutrinos νI and right-handed neutrinos NJ is given by
VIJ = [m
R ]IJ = [m
D]IJM
J (3)
and are important for the phenomenology of the right-handed neutrinos. For generic Dirac
and Majorana neutrino masses no simple relation exists between the physical masses, left-
right mixing angles and the PMNS matrix. An estimate for the neutrino couplings is
fI ≃ 7× 10−7
0.5eV
30GeV
. (4)
where λν = URfUL has been expressed in terms of two unitary matrices UL/R and a
diagonal matrix f with elements fI . In general UL 6= UPMNS. Similarly, an approximate
relation for the left-right mixing angles is
VIJ ≃
[UPMNS ]JI = 4× 10−6
0.5eV
30GeV
[UPMNS]JI (5)
which is valid for approximately universal right-handed neutrino masses MI ≃ M and
UR ≃ 1. I note that these formulae for the masses and mixing angles are exact in the
limit of universal Majorana masses and no CP violation in the Dirac masses [11]. For
these fiducial values of the parameters no limits exist from the neutrinoless double β de-
cay experiments or collider searches [5] because the mixing angles are too tiny. No limits
from cosmology exist either since the right-handed neutrinos decay before big bang nucle-
osynthesis if MI∼>O(GeV), which will be assumed throughout (see Section 5 for the decay
length of the right-handed neutrinos).
If a right-handed neutrino is lighter than the Higgs boson, MI < mh, where mh is the
mass of the Higgs boson, then in principle there may be new decay channels
h → NI +X (6)
where X may be a Standard Model particle or another right-handed neutrino (in the latter
case MI +MJ < mh). For instance, from the neutrino coupling one has h → NIνL. This
decay is irrelevant, however, for practical purposes since the rate is too small.
But if it is assumed that at the TeV scale there are new dynamics responsible for
generating both the Higgs boson mass and the right-handed neutrino masses, then higher-
dimension operators involving the two particles should exist and be suppressed by the TeV
scale. These can be a source of new and relevant decay processes. Consider then
δLeff =
O(5)i +
O(6)i + · · ·+ h.c. (7)
where Λ ≃ O(TeV). Only dimension 5 operators are considered here, with dimension
6 operators discussed elsewhere [19]. The central dot ‘·’ denotes a contraction of flavor
indices.
At dimension 5 there are several operators involving right-handed neutrinos. However it
is shown below that constraints from the observed scale of the left-handed neutrino masses
implies that only one of them can be relevant. It is
O(5)1 = H
†HNN (8)
where the flavor dependence is suppressed. The important point is that this operator is
not necessarily suppressed by any small Yukawa couplings. After electroweak symmetry
breaking the only effect of this operator at tree-level is to shift the masses of the right-
handed neutrinos. Constraints on this operator are therefore weak (see below).
This operator, however, can have a significant effect on the Higgs boson. For if
MI +MJ < mh , (9)
the decay
h → NINJ (10)
can occur. For instance, if only a single flavor is lighter than the Higgs boson, the decay
rate is
Γ(h → NINI) =
2β2I + (Imc
where only half the phase space has been integrated over, c
1 /Λ is the coefficient of (8),
and βI ≡ (1− 4M2I /m2h)1/2 is the velocity of the right-handed neutrino.
The dependence of the decay rate on β may be understood from the following comments.
The uninterested reader may skip this paragraph, since this particular dependence is only
briefly referred to later in the next paragraph, and is not particularly crucial to any other
discussion. Imagine a scattering experiment producing the two Majorana fermions only
through an on-shell Higgs boson in the s-channel. The cross-section for this process is
related to the decay rate into this channel, and in particular their dependence on the final
state phase space are identical. Conservation of angular momentum, and when appropriate,
conservation of CP in the scattering process then fixes the dependence of Γ on phase
space. For example, note that the phase of c
1 is physical and cannot be rotated away.
When Imc
1 = 0 the operator (8) conserves CP and the decay rate has the β
3 dependence
typical for fermions. This dependence follows from the usual argument applied to Majorana
fermions: a pair of Majorana fermions has an intrinsic CP parity of −1 [20], so conservation
of CP and total angular momentum in the scattering process implies that the partial wave
amplitude for the two fermions must be a relative p-wave state. If the phase of c
non-vanishing, then CP is broken and the partial wave amplitude can have both p-wave
and s-wave states while still conserving total angular momentum. The latter amplitude
leads to only a βI phase space suppression.
There is a large region of parameter space where this decay rate is larger than the
rate for the Higgs boson to decay into bottom quarks, and, if kinematically allowed, not
significantly smaller than the rate for the Higgs boson to decay into electroweak gauge
bosons. For example, with Im(c
1 ) = 0 and no sum over I,
Γ(h → NINI)
Γ(h → bb)
β3I (12)
This ratio is larger than 1 for Λ∼<12|c
I TeV . If all three right-handed neutrinos are
lighter than the Higgs boson, then the total rate into these channels is larger than the
rate into bottom quarks for Λ∼<20|c
I TeV. If Im(c
1 ) 6= 0 the operator violates CP
and the region of parameter space where decays to right-handed neutrinos dominate over
decays to bottom quarks becomes larger, simply because now the decay rate has less of a
phase space suppression, as described above. The reason for the sensitivity to large values
of Λ is because the bottom Yukawa coupling is small. For
mh > 2mW (13)
the Higgs boson can decay into a pair of W bosons with a large rate and if kinematically
allowed, into a pair of Z gauge bosons with a branching ratio of approximately 1/3. One
finds that with Im(c
1 ) = 0 and no sum over I,
Γ(h → NINI)
Γ(h → WW )
f(βW )
where f(βW ) = 3/4 − β2W /2 + 3β4W /4 [21] and βW is the velocity of the W boson. Still,
the decay of the Higgs boson into right-handed neutrinos is not insignificant. For example,
with Λ ≃ 2 TeV, c(5)1 = 1 and βI ≃ 1, the branching ratio for a Higgs boson of mass 300
GeV to decay into a single right-handed neutrino flavor of mass 30 GeV is approximately
5%. Whether the decays of the Higgs boson into right-handed neutrinos are visible or not
depends on the lifetime of the right-handed neutrino. That issue is discussed in Section 5.
It is now shown that all the other operators at d = 5 involving right-handed neutrinos
and Higgs bosons are irrelevant for the decay of the Higgs boson. Aside from (8), there is
only one more linearly independent operator involving the Higgs boson and a neutrino,
O(5)2 = LH̃LH̃ . (15)
After electroweak symmetry breaking this operator contributes to the left-handed neutrino
masses, so its coefficient must be tiny, c
2/Λ∼<O(mνL) . Consequently, the decay of the
Higgs boson into active neutrinos from this operator is irrelevant. In Section 4 it is seen
that under the minimal flavor violation hypothesis this operator is naturally suppressed
to easily satisfy the condition above. It is then consistent to assume that the dominant
contribution to the active neutrino masses comes from mass mixing with the right-handed
neutrinos.
Other operators involving the Higgs boson exist at dimension 5, but all of them can be
reduced to (15) and dimension 4 operators by using the equations of motion. For instance,
O(5)3 ≡ −i(∂
µN)σµLH̃ → mRNLH̃ + (H̃L)λTν (LH̃) , (16)
where the equations of motion were used in the last step. As a result, this operator does
not introduce any new dynamics. Still, its coefficients must be tiny enough to not generate
too large of a neutrino mass. In particular, enough suppression occurs if its coefficients
are less than or comparable to the neutrino couplings. Under the minimal flavor violation
hypothesis it is seen that these coefficients are naturally suppressed to this level.
Even if the operators O(5)2 and O
3 are not present at tree-level, they will be generated
at the loop-level through operator mixing with O(5)1 . This is because the overall lepton
number symmetry U(1)LN is broken with both the neutrino couplings and O
1 present.
However, such mixing will always involve the neutrino couplings and be small enough to
not generate too large of a neutrino mass. To understand this, it is useful to introduce
a different lepton number under which the right-handed neutrinos are neutral and both
the charged leptons and left-handed neutrinos are charged. Thus the neutrino couplings
and the operators O(5)2 and O
3 violate this symmetry, but the operator O
1 preserves
it. In the limit that λν → 0 this lepton number symmetry is perturbatively exact, so
inserting O(5)1 into loops can only generate O
2 and O
3 with coefficients proportional to
the neutrino couplings. Further, O(5)2 violates this symmetry by two units, so in generating
it from loops of Standard Model particles and insertions of O(5)1 it will be proportional
to at least two powers of the neutrino couplings. Likewise, in generating O(5)3 from such
loops its coefficient is always proportional to at least one power of the neutrino coupling.
In particular, O(5)2 is generated directly at two-loops, with c
2 ∝ λTν λνc
1 . It is also
generated indirectly at one-loop, since O(5)3 is generated at one-loop, with c
3 ∝ c
1 λν .
These operator mixings lead to corrections to the neutrino masses that are suppressed by
loop factors and at least one power of mR/Λ compared to the tree-level result.
As a result, no significant constraint can be applied to the operator O(5)1 .2 Instead the
challenge is to explain why the coefficients of O(5)2 and O
3 in the effective theory are small
to begin with. The preceding arguments show why it is technically natural for them to
be small, even if O(5)1 is present. The minimal flavor violation hypothesis discussed below
does provide a technically consistent framework in which this occurs.
3 Naturalness
The operator
H†HNN (17)
violates chirality, so it contributes to the mass of the right-handed neutrino at both tree
and loop level. At tree level
δmR = c
= 60c
GeV . (18)
There is also a one-loop diagram with an insertion of this operator. It has a quadratic
divergence such that
δmR ≃ 2c
. (19)
Similarly, at one-loop
δm2h ≃
1 mR]Λ . (20)
2This statement assumes c(5)∼<O(16π
2) and that the loop momentum cutoff Λloop ≃ Λ. Constraints
might conceivably occur for very light right-handed neutrino masses, but that possibility is not explored
here since MI∼>O(GeV) is assumed throughout in order that the right-handed neutrinos decay before big
bang nucleosynthesis.
1 ∼ O(1) then a right-handed neutrino with mass MI ≃ 30 GeV requires O(1) tuning
for TeV∼< Λ∼< 10 TeV, and mh ≃ 100 GeV is technically natural unless Λ∼> 10 TeV or mR
is much larger than the range (MI∼<150 GeV ) considered here.
Clearly, if Λ ∼> O(10 TeV) then a symmetry would be required to protect the right-
handed neutrino and Higgs boson masses. One such example is supersymmetry. Then
this operator can be generalized to involve both Higgs superfields and would appear in the
superpotential. It would then be technically natural for the Higgs boson and right-handed
neutrino masses to be protected, even for large values of Λ. As discussed previously, for
such large values of Λ decays of the Higgs boson into right-handed neutrinos may still be
of phenomenological interest.
4 Minimal Flavor Violation
The higher dimension operators involving right-handed neutrinos and Standard Model
leptons previously discussed can a priori have an arbitrary flavor structure and size. But
as is well-known, higher dimension operators in the lepton and quark sector suppressed
by only Λ ≃ TeV −10 TeV are grossly excluded by a host of searches for flavor changing
neutral currents and overall lepton number violating decays.
A predictive framework for the flavor structure of these operators is provided by the
minimal flavor violation hypothesis [9, 10, 11]. This hypothesis postulates a flavor symme-
try assumed to be broken by a minimal set of non-dynamical fields, whose vevs determine
the renormalizable Yukawa couplings and masses that violate the flavor symmetry. Since a
minimal field content is assumed, the flavor violation in higher dimension operators is com-
pletely determined by the now irreducible flavor violation appearing in the right-handed
neutrino masses and the neutrino, charged lepton and quark Yukawa couplings. Without
the assumption of a minimal field content breaking the flavor symmetries, unacceptably
large flavor violating four fermion operators occur. In practice, the flavor properties of a
higher dimension operator is determined by inserting and contracting appropriate powers
and combinations of Yukawa couplings to make the operator formally invariant under the
flavor group. Limits on operators in the quark sector are 5− 10 TeV [10], but weak in the
lepton sector unless the neutrinos couplings are not much less than order unity [11][22].
It is important to determine what this principle implies for the size and flavor structure
of the operator
1 )IJH
†HNINJ . (21)
It is seen below that the size of its coefficients depends critically on the choice of the flavor
group for the right-handed neutrinos. This has important physical consequences which are
then discussed.
In addition one would like to determine whether the operators O(5)2 and O
3 are suf-
ficiently suppressed such that their contribution to the neutrinos masses is always sub-
dominant. In Section 2 it was argued that if these operators are initially absent, radiative
corrections involving O(5)1 and the neutrino couplings will never generate large coefficients
(in the sense used above) for these operators. However, a separate argument is needed to
explain why they are initially small to begin with. It is seen below that this is always the
case assuming minimal flavor violation.
To determine the flavor structure of the higher dimension operators using the minimal
flavor violation hypothesis, the transformation properties of the particles and couplings are
first defined. The flavor symmetry in the lepton sector is taken to be
GN × SU(3)L × SU(3)ec × U(1) (22)
where U(1) is the usual overall lepton number acting on the Standard Model leptons. With
right-handed neutrinos present there is an ambiguity over what flavor group to choose for
the right-handed neutrinos, and what charge to assign them under the U(1). In fact, since
there is always an overall lepton number symmetry unless both the Majorana masses and
the neutrino couplings are non-vanishing, there is a maximum of two such U(1) symmetries.
Two possibilities are considered for the flavor group of the right-handed neutrinos:
GN = SU(3)× U(1)′ or SO(3) . (23)
The former choice corresponds to the maximal flavor group, whereas the latter is chosen to
allow for a large coupling for the operator (21), shown below. The fields transform under
the flavor group SU(3)× SU(3)L × SU(3)ec × U(1)′ × U(1) as
N → (3,1,1)(1,0) (24)
L → (1,3,1)(−1,1) (25)
ec → (1,1,3)(1,−1) , (26)
Thus U(1)′ is a lepton number acting on the right-handed neutrinos and Standard Model
leptons and is broken only by the Majorana masses. U(1) is a lepton number acting only
on the Standard Model leptons and is only broken by the neutrino couplings. Then the
masses and Yukawa couplings of the theory are promoted to spurions transforming under
the flavor symmetry. Their representations are chosen in order that the Lagrangian is
formally invariant under the flavor group. Again for GN = SU(3) × U(1)′,
λν → (3,3,1)(0,−1) (27)
λl → (1,3,3)(0,0) (28)
mR → (6,1,1)(−2,0) . (29)
For GN = SO(3) there are several differences. First, the 3’s of SU(3) simply become 3’s of
SO(3). Next, the U(1) charge assignments remain but there is no U(1)′ symmetry. Finally,
a minimal field content is assumed throughout, implying that for GN = SO(3) mR ∼ 6 is
real.
With these charge assignments a spurion analysis can now be done to estimate the size
of the coefficents of the dimension 5 operators introduced in Section 2.
For either choice of GN one finds the following. An operator that violates the U(1)
lepton number by n units is suppressed by n factors of the tiny neutrino couplings. In
particular, the dangerous dimension 5 operators O(5)2 and O
3 are seen to appear with
two and one neutrino couplings, which is enough to suppress their contributions to the
neutrino masses. If GN = SO(3) such operators can also be made invariant under SO(3)
by appropriate contractions. If however GN = SU(3)×U(1)′, then additional suppressions
occur in order to construct GN invariants. For example, the coefficients of the dimension
5 operators O(5)2 and O
3 are at leading order λ
Rλν/Λ and λνm
R/Λ respectively and
are sufficiently small.
It is now seen that the flavor structure of the operator (8) depends on the choice of the
flavor group GN . One finds
GN = SU(3)× U(1)′ : c
1 ∼ a1
mRTr[m
+ · · ·
GN = SO(3) : c
1 ∼ 1+ d1
mR ·mR
+ · · ·+ e1λνλ†ν + · · · (30)
where · · · denotes higher powers in mR and λνλ†ν . Comparing the expressions in (30), the
only important difference between the two is that 1 is invariant under SO(3), but not
under SU(3) or U(1)′. As we shall see shortly, this is a key difference that has important
consequences for the decay rate of the Higgs boson into right-handed neutrinos.
Next the physical consequences of the choice of flavor group are determined. First note
that if we neglect the λνλ
ν ∝ mL contribution to c
1 , then for either choice of flavor group
the right-handed neutrino masses mR and couplings c
1 are simultaneously diagonalizable.
For GN = SO(3) this follows from the assumption that mR ∼ 6 is a real representation.
As a result, the couplings c
1 are flavor-diagonal in the right-handed neutrino mass basis.
If GN = SO(3) the couplings c
1 are flavor-diagonal, universal at leading order, and
not suppressed by any Yukawa couplings. It follows that
Br(h → NINI)
Br(h → NJNJ)
≃ 1 (31)
up to small flavor-diagonal corrections of order mR/Λ from the next-to-leading-order terms
in the couplings c
1 . βI is the velocity of NI and its appearance in the above ratio is simply
from phase space. It is worth stressing that even if the right-handed neutrino masses are
non-universal, the branching ratios of the Higgs boson into the right-handed neutrinos are
approximately universal and equal to 1/3 up to phase space corrections. The calculations
from Section 2 of the Higgs boson decay rate into right-handed neutrinos do not need to be
rescaled by any small coupling, and the conclusion that these decay channels dominate over
h → bb for Λ up to 20 TeV still holds. Theoretically though, the challenge is to understand
why MI ≪ Λ.
Similarly, if GN = SU(3) the couplings are flavor-diagonal and suppressed by at least
a factor of mR/Λ but not by any Yukawa couplings. This suppression has two effects.
First, it eliminates the naturalness constraints discussed in Section 3. The other is that it
suppresses the decay rate of h → NINI by a predictable amount. In particular
Γ(h → NINI) =
I (32)
where I have set a1 = 1, and
Br(h → NINI)
Br(h → NJNJ)
up to flavor-diagonal corrections of order mR/Λ. In this case, the Higgs boson decays
preferentially to the right-handed neutrino that is the heaviest. Still, even with this sup-
pression these decays dominate over h → bb up to Λ ≃ 1 TeV if three flavors of right-handed
neutrinos of mass MI ≃ O(50GeV) are lighter than the Higgs boson. For larger values of
Λ these decays have a subdominant branching fraction. They are still interesting though,
because they have a rich collider phenomenology and may still be an important channel
in which to search for the Higgs boson. This scenario might be more natural theoretically,
since an approximate SU(3) symmetry is protecting the mass of the fermions.
5 Right-handed Neutrino Decays
I have discussed how the presence of a new operator at the TeV scale can introduce new
decay modes of the Higgs boson into lighter right-handed neutrinos, and described the
circumstances under which these new processes may be the dominant decay mode of the
Higgs boson. In the previous section we have seen that whether that actually occurs or
not depends critically on a few assumptions. In particular, on whether the Higgs boson
is light, on the scale of the new operator, and key assumptions about the identity of the
broken flavor symmetry of the right-handed neutrinos.
Whether the decays of the Higgs boson into right-handed neutrinos are visible or not
depends on the lifetime of the right-handed neutrinos. It is seen below that in the mini-
mal flavor violation hypothesis their decays modes are determined by their renormalizable
couplings to the electroweak neutrinos and leptons, rather than through higher-dimension
operators.
The dominant decay of a right-handed neutrinos is due to the gauge interactions with
the electroweak gauge bosons it acquires through mass mixing with the left-handed neu-
trinos. At leading order a right-handed neutrino NJ acquires couplings to WlI and ZνI
which are identical to those of a left-handed neutrino, except that they are suppressed by
the mixing angles
VIJ = [m
D]IJM
J . (34)
If the right-handed neutrino is heavier than the electroweak gauge bosons but lighter
than the Higgs boson, it can decay as NJ → W+l−I and NJ → ZνI . Since it is a Majorana
particle, decays to charge conjugated final states also occur. The rate for these decays is
proportional to |VIJ |2M3J .
If a right-handed neutrino is lighter than the electroweak gauge bosons, it decays
through an off-shell gauge boson to a three-body final state. Its lifetime can be obtained by
comparing it to the leptonic decay of the τ lepton, but after correcting for some additional
differences described below. The total decay rate is 3
Γtotal(NI)
Γ(τ → µνµντ )
= 2× 9 (cW + 0.40cZ )
. (35)
The corrections are the following. The factor of “9” counts the number of decays available
to the right-handed neutrino through charged current exchange, assuming it to be heavier
than roughly few-10 GeV. The factor of “0.40” counts the neutral current contribution. It
represents about 30% of the branching ratio, with the remaining 70% of the decays through
the charged current. The factor of “2” is because the right-handed neutrino is a Majorana
particle, so it can decay to both particle and anti-particle, e.g. W ∗l− and W ∗l+, or Z∗ν
and Z∗ν. Another correction is due to the finite momentum transfer in the electroweak
gauge boson propagators. This effect is described by the factors cW and cW where
cG(xG, yG) = 2
dzz2(3− 2z)
(1− (1− z)xG)2 + yG
where xG = M
G, yG = Γ
G, cG(0, 0) = 1 and each propagator has been ap-
proximated by the relativistic Breit-Wigner form. The non-vanishing momentum transfer
enhances the decay rate by approximately 10% for mR masses around 30GeV and by ap-
proximately 50% for masses around 50 GeV. This effect primarily affects the overall rate
and is less important to the individual branching ratios.
The formula (36) is also valid when the right-handed neutrino is more massive than
the electroweak gauge bosons such that the previously mentioned on-shell decays occur. In
that case (35) gives the inclusive decay rate of a right-handed neutrino into any electroweak
gauge boson and a charged lepton or a left-handed neutrino. In this case the correction
from the momentum transfer is obviously important to include! It enhances the decay rate
by approximately a factor of 40 for masses around 100 GeV, but eventually scales as M−2I
for a large enough mass.
3An ≈ 2 error in an earlier version has been corrected.
An effect not included in the decay rate formula above is the quantum interference that
occurs in the same flavor l+l−ν or ννν final states. Its largest significance is in affecting the
branching ratio of these specific, subdominant decay channels and is presented elsewhere
[19]. Using cττ = 87µm [23] and BR(→ µνµντ ) = 0.174 [23], (35) gives the following decay
length for NI ,
cτI = 0.90m
cW + 0.40cZ
30 GeV
(120 keV)2
. (37)
Care must be used in interpreting this formula, since the Dirac and Majorana masses
are not completely independent because they must combine together to give the observed
values of the active neutrino masses.
This expression is both model-independent and model-dependent. Up to this point no
assumptions have been made about the elements of the Dirac mass matrix or the right-
handed neutrino masses, so the result above is completely general. Yet the actual value
of the decay length clearly depends on the flavor structure of the Dirac mass matrix. In
particular, the matrix elements [mDm
D]II/MI are not the same as the active neutrino
mass masses. This is fortunate, since it presents an opportunity to measure a different set
of neutrino parameters from those measured in neutrino oscillations.
The masses MI describe 3 real parameters, and a priori the Dirac matrix mD describes
18 real parameters. However, 3 of the phases in mD can be removed by individual lepton
number phase rotations on the left-handed neutrinos and charged leptons, leaving 15 pa-
rameters which I can think of as 6 mixing angles, 3 real Yukawa couplings and 6 phases.
Including the three right-handed neutrino masses gives 18 parameters in total. Five con-
straints on combinations of these 18 parameters already exist from neutrino oscillation
experiments. In principle all of these parameters could be measured through detailed stud-
ies of right-handed neutrino decays, since amplitudes for individual decays are proportional
to the Dirac neutrino matrix elements. However, at tree-level these observables depend
only on |[mD]IJ | and are therefore insensitive to the 6 phases. So by studying tree-level
processes only the 3 right-handed neutrino masses, 3 Yukawa couplings, and 6 mixing
angles could be measured in principle.
In particular, the dominant decay is h → NINI → qqqqlJ lK which contains no miss-
ing energy. Since the secondary events are highly displaced, there should be no confusion
about which jets to combine with which charged leptons. In principle a measurement of
the mass of the right-handed neutrino and the Higgs boson is possible by combining the in-
variant momentum in each event. A subsequent measurement of a right-handed neutrino’s
lifetime from the spatial distribution of its decays measures [mDm
D]II . More information
is acquired by measuring the nine branching ratios BR(NI → qq′lJ) ∝ |[mD]IJ |2. Such
measurements provide 6 additional independent constraints. In total, 12 independent con-
straints on the 18 parameters could in principle be obtained from studying right-handed
neutrino decays at tree-level.
To say anything more precise about the decay length would require a model of the
neutrino couplings and right-handed neutrino mass parameters. Specific predictions could
be done within the context of such a model. Of interest would be the branching ratios and
the mean and relative decay lengths of the three right-handed neutrinos.
The factor [mDm
D]II/MI appearing in the decay length is not the active neutrino mass
obtained by diagonalizing mTDm
R mD, but it is close. If I approximate [mDm
D]II/MI ≃
mI , then
cτI ≃ 0.90m
30 GeV
0.48 eV
cW + 0.4cZ
A few comments are in order. First, the decay lengths are macroscopic, since by inspection
they range from O(100µm) to O(10m) for a range of parameters, and for these values are
therefore visible at colliders. Next, the decay length is evidently extremely sensitive to MI .
Larger values of MI have shorter decays lengths. For instance, if MI = 100 GeV (which
requiresmh > 200 GeV) andmI = 0.5 eV then cτI ≃ 0.2mm. Finally, if the active neutrino
masses are hierarchical, then one would expect M4I cτI to be hierarchical as well, since this
quantity is approximately proportional to m−1L . One or two right-handed neutrinos may
therefore escape the detector if the masses of the lightest two active neutrinos are small
enough.
I have described decays of the right-handed neutrinos caused by its couplings to elec-
troweak gauge bosons acquired through mass mixing with the left-handed neutrinos. How-
ever, additional decay channels occur through exchange of an off-shell Higgs boson, higher
dimension operators or loop effects generated from its gauge couplings. It turns out that
these processes are subdominant, but may be of interest in searching for the Higgs bo-
son. Exchange of an off-shell Higgs boson causes a decay NI → νJbb which is suppressed
compared to the charged and neutral current decays by the tiny bottom Yukawa coupling.
Similarly, the dimension 5 operator (8) with generic flavor couplings allows for the decay
NI → NJh∗ → NJbb for NJ lighter than NI 4. However, using the minimal flavor vio-
lation hypothesis it was shown in Section 4 that the couplings of that higher dimension
operator are diagonal in the same basis as the right-handed neutrino mass basis, up to
flavor-violating corrections that are at best O(λ2ν) (see (30)). As result, this decay is highly
suppressed. At dimension 5 there is one more operator that I have not yet introduced
which is the magnetic moment operator
·NσρσNBρσ (39)
and it involves only two right-handed neutrinos. It causes a heavier right-handed neutrino
to decay into a lighter one, NI → NJ+γ/Z for I 6= J . To estimate the size of this operator,
first note that its coefficient must be anti-symmetric in flavor. Then in the context of
minimal flavor violation with GR = SO(3), the leading order term is c
4 ≃ [λνλ
ν ]AS where
4The author thanks Scott Thomas for this observation.
“AS” denotes ‘anti-symmetric part’. This vanishes unless the neutrino couplings violate
CP . In that case the amplitude for this decay is of order (λν)
2. If GR = SU(3)×U(1)′ the
leading order term cannot be [mR]AS(Tr[mRm
q)n/Λn+q, since they vanish in the right-
handed neutrino mass basis. The next order involves λνλ
ν and some number of mR’s,
but there does not appear to be any invariant term. Thus for either choice of GR the
amplitude for NI decays from this operator are O(λ
ν) or smaller, which is much tinier
than the amplitudes for the other right-handed neutrino decays already discussed which
are of order λν . Subdominant decays N → ν + γ can occur from dimension 6 operators
and also at also one-loop from electroweak interactions, but in both cases the branching
ratio is tiny [19].
6 Discussion
In order for these new decays to occur at all requires that the right-handed neutrinos are
lighter than the Higgs boson. But from a model building perspective, one may wonder
why the right-handed neutrinos are not heavier than the scale Λ. A scenario in which
the right-handed neutrinos are composite would naturally explain why these fermions are
comparable or lighter than the compositeness scale Λ, assumed to be O(TeV). Since their
interactions with the Higgs boson through the dimension 5 operator (8) are not small, the
Higgs boson would be composite as well (but presumed to be light).
These new decay channels of the Higgs boson will be the dominant decay modes if
the right-handed neutrinos are also lighter than the electroweak gauge bosons, and if the
coefficient of the higher dimension operator (8) is not too small. As discussed in Section
4, in the minimal flavor violation framework the predicted size of this operator depends
on the choice of approximate flavor symmetries of the right-handed neutrinos. It may be
O(1) or O(mR/Λ).
In the former situation the new decays dominate over Higgs boson decays to bottom
quarks for scales Λ∼<10 − 20 TeV, although only scales Λ ≃ 1 − 10 TeV are technically
natural. This case however presents a challenge to model building, since the operator (8)
breaks the chirality of the right-handed neutrinos. Although it may be technically natural
for the right-handed neutrinos to be much lighter than the scale Λ (see Section 3), one
might expect that any theory which generates a large coefficient for this operator to also
generate Majorana masses mR ∼ O(Λ).
In the case where the coefficient of (8) is O(mR/Λ) the new decays can still dominate
over decays to bottom quarks provided that the scale Λ ≃ O(1 TeV). For larger values of
Λ these decays are subdominant but have sizable branching fractions up to Λ ≃ O(10TeV).
This situation might be more amendable to model building. For here an approximate
SU(3) symmetry is protecting the mass of the right-handed neutrinos.
In either case though one needs to understand why the right-handed neutrinos are para-
metrically lighter than Λ. It would be extremely interesting to find non-QCD-type theories
of strong dynamics where fermions with masses parametrically lighter than the scale of
strong dynamics occur. Or using the AdS/CFT correspondence [8], to find a Randall-
Sundrum type model [7] that engineers this outcome. The attitude adopted here has been
to assume that such an accident or feature can occur and to explore the consequences.
Assuming that these theoretical concerns can be naturally addresseed, the Higgs boson
physics is quite rich. To summarize, in the new process the Higgs boson decays through a
cascade into a six-body or four-body final state depending on the masses of the right-handed
neutrinos. First, it promptly decays into a pair of right-handed neutrinos, which have a
macroscopic decay length anywhere from O(100µm − 10m) depending on the parameters
of the Majorana and Dirac neutrino masses. If one or two active neutrinos are very light,
then the decay lengths could be larger. Decays occurring in the detector appear as a pair of
displaced vertices. For most of the time each secondary vertex produces a quark pair and a
charged lepton, dramatically violating lepton number. For a smaller fraction of the time a
secondary vertex produces a pair of charged leptons or a pair of quarks, each accompanied
with missing energy. From studying these decays one learns more about neutrinos and
the Higgs boson, even if these channels should not form the dominant decay mode of the
Higgs boson. The experimental constraints on this scenario from existing colliders and its
discovery potential at the LHC will be described elsewhere [19] [24].
Acknowledgments
The author thanks Matt Stassler and Scott Thomas for discussions. This work is supported
by the U.S. Department of Energy under contract No. DE-FG03-92ER40689.
References
[1] M. Fukugita and T. Yanagida, Phys. Lett. B 174, 45 (1986).
[2] T. Asaka and M. Shaposhnikov, Phys. Lett. B 620, 17 (2005) [arXiv:hep-ph/0505013];
M. Shaposhnikov, J. Phys. Conf. Ser. 39, 9 (2006).
[3] A. Pilaftsis, Z. Phys. C 55, 275 (1992) [arXiv:hep-ph/9901206].
[4] K. Belotsky, D. Fargion, M. Khlopov, R. Konoplich and K. Shibaev, Phys. Rev. D 68,
054027 (2003) [arXiv:hep-ph/0210153].
[5] T. Han and B. Zhang, Phys. Rev. Lett. 97, 171804 (2006) [arXiv:hep-ph/0604064].
[6] P. Q. Hung, arXiv:hep-ph/0612004.
[7] L. Randall and R. Sundrum, Phys. Rev. Lett. 83, 3370 (1999) [arXiv:hep-ph/9905221].
http://arxiv.org/abs/hep-ph/0505013
http://arxiv.org/abs/hep-ph/9901206
http://arxiv.org/abs/hep-ph/0210153
http://arxiv.org/abs/hep-ph/0604064
http://arxiv.org/abs/hep-ph/0612004
http://arxiv.org/abs/hep-ph/9905221
[8] J. M. Maldacena, Adv. Theor. Math. Phys. 2, 231 (1998) [Int. J. Theor. Phys. 38,
1113 (1999)] [arXiv:hep-th/9711200]; S. S. Gubser, I. R. Klebanov and A. M. Polyakov,
Phys. Lett. B 428, 105 (1998) [arXiv:hep-th/9802109]; E. Witten, Adv. Theor. Math.
Phys. 2, 253 (1998) [arXiv:hep-th/9802150].
[9] R. S. Chivukula and H. Georgi, Phys. Lett. B 188, 99 (1987);
[10] G. D’Ambrosio, G. F. Giudice, G. Isidori and A. Strumia, Nucl. Phys. B 645, 155
(2002) [arXiv:hep-ph/0207036].
[11] V. Cirigliano, B. Grinstein, G. Isidori and M. B. Wise, Nucl. Phys. B 728, 121 (2005)
[arXiv:hep-ph/0507001].
[12] S. Dimopoulos, M. Dine, S. Raby and S. D. Thomas, Phys. Rev. Lett. 76, 3494
(1996) [arXiv:hep-ph/9601367]; S. Dimopoulos, M. Dine, S. Raby, S. D. Thomas
and J. D. Wells, Nucl. Phys. Proc. Suppl. 52A, 38 (1997) [arXiv:hep-ph/9607450];
S. Dimopoulos, S. D. Thomas and J. D. Wells, Nucl. Phys. B 488, 39 (1997)
[arXiv:hep-ph/9609434].
[13] M. J. Strassler and K. M. Zurek, arXiv:hep-ph/0604261; M. J. Strassler and
K. M. Zurek, arXiv:hep-ph/0605193; M. J. Strassler, arXiv:hep-ph/0607160.
[14] R. Dermisek and J. F. Gunion, Phys. Rev. Lett. 95, 041801 (2005)
[arXiv:hep-ph/0502105]; R. Dermisek and J. F. Gunion, Phys. Rev. D 73, 111701
(2006) [arXiv:hep-ph/0510322]; S. Chang, P. J. Fox and N. Weiner, JHEP
0608, 068 (2006) [arXiv:hep-ph/0511250]; S. Chang, P. J. Fox and N. Weiner,
[arXiv:hep-ph/0608310].
[15] L. M. Carpenter, D. E. Kaplan and E. J. Rhee, arXiv:hep-ph/0607204.
[16] A. V. Manohar and M. B.Wise, Phys. Lett. B 636, 107 (2006) [arXiv:hep-ph/0601212].
[17] A. V. Manohar and M. B. Wise, Phys. Rev. D 74, 035009 (2006)
[arXiv:hep-ph/0606172].
[18] B. Pontecorvo, Sov. Phys. JETP 6, 429 (1957) [Zh. Eksp. Teor. Fiz. 33, 549 (1957)].
ibid. Sov. Phys. JETP 7, 172 (1958) [Zh. Eksp. Teor. Fiz. 34, 247 (1957)]; Z. Maki,
M. Nakagawa and S. Sakata, Prog. Theor. Phys. 28, 870 (1962).
[19] M. L. Graesser, arXiv:0705.2190 [hep-ph].
[20] S. Weinberg, “The Quantum theory of fields. Vol. 1: Foundations,” Cambridge, UK:
Univ. Pr. (1995) 609 p
[21] J. F. Gunion, H. E. Haber, G. L. Kane and S. Dawson, The Higgs Hunter’s Guide;
ibid., [arXiv:hep-ph/9302272].
http://arxiv.org/abs/hep-th/9711200
http://arxiv.org/abs/hep-th/9802109
http://arxiv.org/abs/hep-th/9802150
http://arxiv.org/abs/hep-ph/0207036
http://arxiv.org/abs/hep-ph/0507001
http://arxiv.org/abs/hep-ph/9601367
http://arxiv.org/abs/hep-ph/9607450
http://arxiv.org/abs/hep-ph/9609434
http://arxiv.org/abs/hep-ph/0604261
http://arxiv.org/abs/hep-ph/0605193
http://arxiv.org/abs/hep-ph/0607160
http://arxiv.org/abs/hep-ph/0502105
http://arxiv.org/abs/hep-ph/0510322
http://arxiv.org/abs/hep-ph/0511250
http://arxiv.org/abs/hep-ph/0608310
http://arxiv.org/abs/hep-ph/0607204
http://arxiv.org/abs/hep-ph/0601212
http://arxiv.org/abs/hep-ph/0606172
http://arxiv.org/abs/0705.2190
http://arxiv.org/abs/hep-ph/9302272
[22] V. Cirigliano and B. Grinstein, Nucl. Phys. B 752, 18 (2006) [arXiv:hep-ph/0601111].
[23] W. M. Yao et al. [Particle Data Group], J. Phys. G 33, 1 (2006).
[24] Michael L. Graesser, Amit Lath and Jessie Shelton, work in progress.
http://arxiv.org/abs/hep-ph/0601111
	Motivation
	Higgs Boson Decay
	Naturalness
	Minimal Flavor Violation
	Right-handed Neutrino Decays
	Discussion
