Correlation between the Charged Current Interactions of Light and Heavy Majorana Neutrinos

0.17 MB·English

by  Zhi-zhong Xing

Correlation between the Charged Current Interactions
of Light and Heavy Majorana Neutrinos
Zhi-zhong Xing ∗
Institute of High Energy Physics, Chinese Academy of Sciences, Beijing, China
Abstract
8
0
0
2
The evidence for neutrino oscillations implies that three neutrino flavors

n
(ν ,ν ,ν ) must have different mass states (ν ,ν ,ν ). The most popular
a e µ τ 1 2 3
J idea of generating tiny masses of ν is to introduce three heavy Majorana
i
4 neutrinos N (for i = 1,2,3) into the standard model and implement the
i
2
seesaw mechanism. In this approach the neutrino mixing matrix V appearing

] in the charged current interactions of ν is not unitary, and the strength
h i
p of unitarity violation of V is associated with the matrix R which describes
-
the strength of charged current interactions of N . We present an explicit
p i
e parametrization of the correlation between V and R in terms of nine rotation
h
angles and nine phase angles, which can be measured or constrained in the
[

precisionneutrinooscillationexperimentsandbyexploringpossiblesignatures
4
v of N at the LHC and ILC. Two special but viable scenarios, the Type-I
i
0
seesaw model with two heavy Majorana neutrinos and the Type-II seesaw
2
2 model with one heavy Majorana neutrino and one Higgs triplet, are taken
2 into account to illustrate the simplified V-R correlation. The implications
.
9 of R = 0 on the low-energy neutrino phenomenology are also discussed. In
0 6
particular, we demonstrate that the non-unitarity of V is possible to give rise
7
0 to an appreciable CP-violating asymmetry between ν ν and ν ν
µ τ µ τ
: → →
v oscillations with short or medium baselines.
i
X
PACS number(s): 14.60.Pq, 13.10.+q, 25.30.Pt
r
a
Typeset using REVTEX
∗E-mail: [email protected]
1
1 Veryrobust evidence fornon-zeroneutrino masses andlargeleptonflavor mixing has
recently been achieved from solar [1], atmospheric [2], reactor [3] and accelerator [4] neutrino
oscillation experiments. This great breakthrough opens a new window to physics beyond the
standard model (SM), because the SM itself only contains three massless neutrinos whose
flavor states ν (for α = e,µ,τ) and mass states ν (for i = 1,2,3) are identical. The most
α i
natural andpopular way to generate non-vanishing but tiny masses m for ν is to extend the
i i
SM by introducing three right-handed neutrinos but preserving its SU(2) U(1) gauge
L Y
×
symmetry. This class of neutrino mass models is commonly referred to as the Type-I seesaw
models [5]. If one Higgs triplet ∆ and three right-handed neutrinos are simultaneously
L
introduced into the SM, the resultant SU(2) U(1) gauge models are usually classified as
L Y
×
theType-II seesaw models[6]. Ineither case themass statesof threeright-handedneutrinos,
denoted as N (for i = 1,2,3), have the positive eigenvalues M which can be much larger
i i
thanthe electroweak symmetry breaking scale v 174GeV.The smallness of m is therefore
≈ i
attributed to the smallness of v2/M (and the smallness of the vacuum expectation value of
i
∆ intheType-IIseesawmodels). BothlightandheavyneutrinosaretheMajoranaparticles
L
in such seesaw models, in which the lepton number (L) is not conserved. Associated with
the seesaw mechanism, the leptogenesis mechanism [7] may naturally work to account for
the cosmological matter-antimatter asymmetry via the CP-violating and out-of-equilibrium
decays of N and the (B L)-conserving sphaleron processes [8].
i −
To directly test the seesaw and leptogenesis mechanisms, it is desirable to experimentally
discover the heavy Majorana neutrinos N at high-energy e+e− and (or) hadron colliders.
i
Since such neutral and weakly interacting particles leave no tracein ordinary detectors, their
possible collider signatures must involve charged leptons via the charged current interactions
for some ∆L = 2 processes (i.e., those processes with the lepton number being violated by
two units) [9]. In the basis of mass states, the standard charged current interactions of ν
i
and N can be written as
i
ν N
g 1 1
= (e µ τ) Vγµν  W− +(e µ τ) RγµN  W− + h.c. , (1)
−Lcc √2 L 2 µ L 2 µ
 ν  N  
  3L  3L 
where V is just the Maki-Nakagawa-Sakata (MNS) neutrino mixing matrix [10] responsible
for neutrino oscillations, and R describes the strength of charged current interactions be-
tween (e,µ,τ) and (N ,N ,N ). It has been noticed that V is not exactly unitary in the
1 2 3
seesaw models and its deviation from unitarity is essentially characterized by non-vanishing
R [11]. Indeed, the production and detection of N at the Large Hadron Collider (LHC) or
i
<
the International Linear Collider (ILC) require not only M (10) TeV but also apprecia-
ble sizes of the matrix elements of R. Hence the unitarityiv∼iolOation of V might show up in
the future long-baseline neutrino oscillation experiments, if the collider signatures of heavy
Majorana neutrinos are really accessible. In this sense, the correlation between V and R
signifies an important relationship between neutrino physics and collider physics.
Thepurposeofthisworkisjusttorevealhowthechargedcurrentinteractionsoflightand
heavy Majorana neutrinos arecorrelated with each other. We shallshow that thecorrelation
between V and R can in general be parametrized in terms of nine rotation angles and nine
phase angles. This parametrization is independent of any details of a Type-I or Type-II
2
seesaw model, and its parameters can be measured or constrained in the precision neutrino
oscillation experiments and by exploring possible signatures of N at the LHC and ILC.
i
We shall take two special but viable examples, the Type-I seesaw model with two heavy
Majorana neutrinos and the Type-II seesaw model with one heavy Majorana neutrino and
one Higgs triplet, to illustrate the simplified form of V-R correlation. The implications of
R = 0 on the low-energy neutrino phenomenology will also be discussed. In particular,
6
we shall demonstrate that the non-unitarity of V is possible to give rise to an appreciable
CP-violating asymmetry between ν ν and ν ν oscillations with short or medium
µ → τ µ → τ
baselines.
2 Without loss of generality, we choose the basis in which the flavor and mass states
of three charged leptons are identical. In this basis, the neutrino mass terms generated from
spontaneous SU(2) U(1) U(1) symmetry breaking can be written as
L × Y → em
1 M M νc
= (ν Nc) L D L + h.c. , (2)
−Lmass 2 L R (cid:18)MT M (cid:19)(cid:18)N (cid:19)
D R R
where νc and Nc are defined as νc Cν T and Nc CN T with C being the charge
L R L ≡ L R ≡ R
conjugation matrix, M = Y v and M = Y v result from the Yukawa interactions of
D ν L ∆ L
<
Higgs doublet and triplet with v 174 GeV and v 1 GeV being the corresponding
vacuum expectation values, and M≈ is the mass maLtri∼x of three right-handed Majorana
R
neutrinos. The overall 6 6 neutrino mass matrix in , denoted as , is symmetric
× Lmass M
and can be diagonalized by a unitary transformation † ∗ = ; or explicitly,
U MU M
c
V R † M M V R ∗ M 0
L D = ν , (3)
(cid:18)S U (cid:19) (cid:18)MT M (cid:19)(cid:18)S U (cid:19) (cid:18) 0 M (cid:19)
D R c N
c
whereM = Diag m ,m ,m andM = Diag M ,M ,M withm andM (fori = 1,2,3)
ν { 1 2 3} N { 1 2 3} i i
beingthelightandheavyMajorananeutrinomasses, respectively. Afterthisdiagonalization,
c c
one may express the neutrino flavor states ν (for α = e,µ,τ) in terms of the light and heavy
α
neutrino mass states ν and N (for i = 1,2,3):
i i
ν ν N
e 1 1
ν  = V ν  +RN  . (4)
µ 2 2
ν  ν  N 
 τ L  3L  3L
Then the standard charged current interactions between ν and α (for α = e,µ,τ) in the
α
basis of flavor states,
ν
g e
= (e µ τ) γµν  W− + h.c. , (5)
−Lcc √2 L µ µ
ν 
 τ L
turn out to be of the form given in Eq. (1) in the basis of mass states. It is clear that V and
R describe the charged current interactions of three light and heavy Majorana neutrinos,
respectively. While V can be measured from a variety of neutrino oscillation experiments,
R may be determined from possible collider signatures of N .
i
3
Note that V itself is not unitary. Indeed, VV† + RR† = 1 holds as a consequence of
unitarity of the 6 6 transformation matrix in Eq. (3) [12]. The non-unitarity of the
× U
MNS matrix V is an intrinsic feature of the seesaw models, no matter whether they are of
type I or of type II. Since V and R are two 3 3 sub-matrices of , their elements must be
× U
correlated with each other. To find out the explicit correlation between V and R, we may
parametrize in terms of 15 rotation angles and 15 phase angles [13]. Then the common
U
parameters appearing in both V and R characterize their correlation. First of all, let us
define the 2-dimensional (1,2), (1,3) and (2,3) rotation matrices in a 6-dimensional complex
space:
c sˆ∗ 0 0 0 0
12 12
 sˆ c 0 0 0 0
− 12 12
 0 0 1 0 0 0
O =   ,
12  0 0 0 1 0 0
 
 0 0 0 0 1 0
 
 
 0 0 0 0 0 1
 
c 0 sˆ∗ 0 0 0
13 13
 0 1 0 0 0 0
 sˆ 0 c 0 0 0
O = − 13 13  ,
13  0 0 0 1 0 0
 
 0 0 0 0 1 0
 
 
 0 0 0 0 0 1
 
1 0 0 0 0 0
0 c sˆ∗ 0 0 0
23 23
0 sˆ c 0 0 0
O =  − 23 23  , (6)
23 0 0 0 1 0 0
 
0 0 0 0 1 0
 
 
0 0 0 0 0 1
 
where c cosθ and sˆ eiδij sinθ with θ and δ being the rotation angle and phase
ij ≡ ij ≡ ij ij ij
angle, respectively. Other 2-dimensional rotation matrices O (for 1 i < j 6) can be
ij ≤ ≤
defined in an analogous way [14]. We parametrize the 6 6 unitary matrix as
× U
A R V 0
= 0 (7)
U (cid:18)B U (cid:19)(cid:18) 0 1(cid:19)
with
A R
= O O O O O O O O O O O O ,
(cid:18)B U (cid:19) 56 46 36 26 16 45 35 25 15 34 24 14
V 0
0 = O O O . (8)
(cid:18) 0 1(cid:19) 23 13 12
Comparing between Eqs. (3) and (7), we get V = AV and S = BV , in which
0 0
c c sˆ∗ c sˆ∗
12 13 12 13 13
V =  sˆ c c sˆ sˆ∗ c c sˆ∗ sˆ sˆ∗ c sˆ∗  (9)
0 − 12 23 − 12 13 23 12 23 − 12 13 23 13 23
 sˆ sˆ c sˆ c c sˆ sˆ∗ sˆ c c c 
 12 23 − 12 13 23 − 12 23 − 12 13 23 13 23
4
is just the standard parametrization of the unitary MNS matrix (up to some proper phase
rearrangements) [15]. It is obvious that V V in the limit of A 1 (or equivalently,
→ 0 →
R 0 and S 0). Thus A signifies the unitarity violation of V. After a lengthy but
→ →
straightforward calculation, we obtain the explicit expressions of A and R as follows:
c c c 0 0
14 15 16
 
c c sˆ sˆ∗ c sˆ sˆ∗ c
 − 14 15 16 26 − 14 15 25 26 c c c 0 
 sˆ sˆ∗ c c 24 25 26 
A =  − 14 24 25 26  ,
 c c sˆ c sˆ∗ +c sˆ sˆ∗ sˆ sˆ∗ 
 −c14sˆ15c16sˆ2∗6c36 +sˆ14sˆ1∗5c25sˆ26sˆ∗36 −c24c25sˆ26sˆ∗36 −c24sˆ25sˆ∗35c36 c c c 
 − 14 15 25 35 36 14 24 25 26 36 sˆ sˆ∗ c c 34 35 36
 +sˆ sˆ∗ sˆ sˆ∗ c sˆ c sˆ∗ c c − 24 34 35 36 
 14 24 25 35 36 − 14 24 34 35 36 
sˆ∗ c c sˆ∗ c sˆ∗
14 15 16 15 16 16
 
sˆ∗ c sˆ sˆ∗ sˆ∗ sˆ sˆ∗ c
 − 14 15 16 26 − 14 15 25 26 sˆ∗ sˆ sˆ∗ +c sˆ∗ c c sˆ∗ 
R =  +c14sˆ∗24c25c26 − 15 16 26 15 25 26 16 26  . (10)
 
 sˆ∗ c sˆ c sˆ∗ +sˆ∗ sˆ sˆ∗ sˆ sˆ∗ 
 −sˆ∗14sˆ15c16sˆ2∗6c36 c14sˆ∗15c25sˆ26sˆ∗36 −sˆ∗15sˆ16c26sˆ∗36 −c15sˆ∗25sˆ26sˆ∗36 c c sˆ∗ 
 − 14 15 25 35 36 − 14 24 25 26 36 +c c sˆ∗ c 16 26 36
 c sˆ∗ sˆ sˆ∗ c +c c sˆ∗ c c 15 25 35 36 
 − 14 24 25 35 36 14 24 34 35 36 
One can see that A and R involve the same parameters: nine rotation angles (θ , θ and
i4 i5
θ for i = 1, 2 and 3) and nine phase angles (δ , δ and δ for i = 1, 2 and 3). If all of
i6 i4 i5 i6
them are switched off, we shall be left with R = 0 and A = 1. Nontrivial A is therefore the
bridge between V = AV and R.
0
Considering the fact that the unitarity violation of V must be a small effect (at most at
the 1% level as constrained by current neutrino oscillation data and precision electroweak
data [16]), we may treat A as a perturbation to V . The smallness of θ (for i = 1,2,3 and
0 ij
j = 4,5,6) allows us to make the excellent approximations
1 (s2 +s2 +s2 ) 0 0
2 14 15 16
A = 1 sˆ sˆ∗ +sˆ sˆ∗ +sˆ sˆ∗ 1 (s2 +s2 +s2 ) 0 + (s4 ) ,
− 14 24 15 25 16 26 2 24 25 26 O ij
sˆ sˆ∗ +sˆ sˆ∗ +sˆ sˆ∗ sˆ sˆ∗ +sˆ sˆ∗ +sˆ sˆ∗ 1 (s2 +s2 +s2 )
 14 34 15 35 16 36 24 34 25 35 26 36 2 34 35 36 
sˆ∗ sˆ∗ sˆ∗
14 15 16
R = 0+sˆ∗ sˆ∗ sˆ∗ + (s3 ) , (11)
24 25 26 O ij
sˆ∗ sˆ∗ sˆ∗ 
 34 35 36
from which one can easily check the validity of VV† = AA† = 1 RR† to a good degree
−
of accuracy. Thus the nine mixing angles in Eq. (10) or (11) are all of (10−1) or smaller,
O
such that the unitarity violation of V can maximally be of (10−2). The nine CP-violating
O
phases of A or R are in general not suppressed, however. It is worth remarking that R
∼
(10−3) to (10−1) may lead to appreciable collider signatures of lepton number violation,
O O
if the masses of heavy Majorana neutrinos M are of (102) GeV to (10) TeV. At the
i O O
LHC, for instance, the promising lepton-number-violating processes mediated by N include
i
pp W±W± µ±µ±jj and pp W± µ±N µ±µ±jj, where the ∆L = 2 like-sign
→ → → → →
dilepton production can unambiguously signal the existence of N [9,17,18]. A preliminary
i
analysis made in Ref. [9] has shown that it is possible to probe (RR†) s2 + s2 + s2
µµ ≈ 24 25 26
5
to (10−4) for M 100 GeV and to (10−2) for M 400 GeV at the LHC with an
O i ∼ O i ∼
integrated luminosity 100 fb−1. The sensitivity will in general become worse for much larger
values of the heavy Majorana neutrino masses.
Note that the triangular form of A is a salient feature of our parametrization. Some
straightforward consequences on V = AV can be obtained from Eqs. (9) and (10).
0
V = c c c sˆ∗ holds. Given θ = 0 for V , which might result from certain flavor
• e3 14 15 16 13 13 0
symmetries imposed on M , M and M [12,18], V turns out to vanish.
L D R e3
The ratio V /V = tanθ is completely irrelevant to the parameters appearing in A
• | e2 e1| 12
or R. This result implies that the extraction of θ from the solar neutrino oscillation
12
data is essentially independent of possible unitarity violation of V.
V 2+ V 2+ V 2 = c2 c2 c2 1 (s2 +s2 +s2 ) holds. Hence non-vanishing θ ,
• | e1| | e2| | e3| 14 15 16 ≈ − 14 15 16 14
θ and θ violate the normalization condition of three matrix elements in the first
15 16
row of V. Current experimental data require (s2 +s2 +s2 ) < 1% [16].
14 15 16
∼
m = c2 c2 c2 m (c c )2 +m (sˆ∗ c )2 +m (sˆ∗ )2 holds for the effective mass of
• h iee 14 15 16| 1 12 13 2 12 13 3 13 |
the neutrinoless double-beta decay. The smallness of θ , θ and θ implies that their
14 15 16
effects on m are in practice negligible.
h iee
m = c c c m2c2 c2 +m2s2 c2 +m2s2 holds for the effective mass term of the
• h ie 14 15 16q 1 12 13 2 12 13 3 13
tritium beta decay. The smallness of θ , θ and θ implies that their effects on m
14 15 16 h ie
are also negligible.
Another consequence of the non-unitarity of V is the loss of universality for the Jarlskog
invariants of CP violation [19], Jij Im(V V V∗V∗), where the Greek indices run over
αβ ≡ αi βj αj βi
(e,µ,τ) and the Latin indices run over (1,2,3). The explicit expressions of Jij in terms of
αβ
θ and δ are rather complicated and will be presented elsewhere. But we shall illustrate
ij ij
that the extra CP-violating phases of V are possible to give rise to a significant asymmetry
between ν ν and ν ν oscillations with short or medium baselines in the subsequent
µ → τ µ → τ
section.
3 For the sake of simplicity, here we only consider a special but interesting pattern
of V : tanθ = 1/√2, θ = 0 and θ = π/4 (namely, V takes the well-known tri-
0 12 13 23 0
bimaximal mixing pattern [20]). Then the non-unitary neutrino mixing matrix V = AV
0
can approximate to
2 1 e−iδ12 0
3 3
 q q 
V 1 (1+2X)eiδ12 1 (1 X) 1 e−iδ23 , (12)
≈  −q6 q3 − q2 
 1 (1 2Y +Z)ei(δ12+δ23) 1 (1+Y +Z)eiδ23 1 (1 Z)
q6 − −q3 q2 − 
where
X = (sˆ sˆ∗ +sˆ sˆ∗ +sˆ sˆ∗ )e−iδ12 ,
14 24 15 25 16 26
Y = (sˆ sˆ∗ +sˆ sˆ∗ +sˆ sˆ∗ )e−i(δ12+δ23) ,
14 34 15 35 16 36
Z = (sˆ sˆ∗ +sˆ sˆ∗ +sˆ sˆ∗ )e−iδ23 . (13)
24 34 25 35 26 36
6
The Jarlskog invariants Jij turn out to be J23 = J23 = J31 = J31 = 0, J12 ImX/3,
αβ eµ τe eµ τe eµ ≈
J12 ImY/3, and
τe ≈
J12 (ImX +ImY)/6 ,
µτ ≈
J23 (ImX +ImY +2ImZ)/6 ,
µτ ≈
J31 (ImX +ImY ImZ)/6 . (14)
µτ ≈ −
ji ij ij ij
Note that J = J = J holds as a direct consequence of the definition of J . It
αβ βα − αβ αβ
becomes clear that different Jij may in general have different values in the presence of
αβ
unitarity violation, which can result in some new CP-violating effects in neutrino oscillations
via the phase parameters δ (for k = 1,2,3 and l = 4,5,6) hidden in X, Y and Z.
kl
Taking the steps outlined in Ref. [16], one may easily derive the probabilities of ν ν
α → β
oscillations in vacuum. The result is 1
V 2 V 2 +2 Re V V V∗ V∗ cos∆ 2 Jij sin∆
Xi | αi| | βi| Xi<j (cid:16) αi βj αj βi(cid:17) ij − Xi<j αβ ij
P(ν ν ) = , (15)
α → β VV† VV†
(cid:16) (cid:17)αα(cid:16) (cid:17)ββ
where ∆ ∆m2 L/(2E) with ∆m2 m2 m2, E being the neutrino beam energy and L
ij ≡ ij ij ≡ i − j
being the baseline length. It is straightforward to write out the expression of P(ν ν )
α → β
from Eq. (15) by making the replacement V = V∗ or equivalently Jij = Jij . If V is
⇒ αβ ⇒ − αβ
exactly unitary (i.e., A = 1 and V = V ), the denominator of Eq. (15) will become unity
0
and the conventional formula of P(ν ν ) will be reproduced. It has been observed in
α → β
Ref. [21] that ν ν and ν ν oscillations may serve as an excellent tool to probe
µ → τ µ → τ
possible signatures of CP violation induced by the non-unitarity of V. To see this point
more clearly, we consider a short- or medium-baseline neutrino oscillation experiment with
sin∆ sin∆ sin∆ , in which the terrestrial matter effects are expected to be
| 13| ∼ | 23| ≫ | 12|
insignificant or negligibly small. Then the dominant CP-conserving and CP-violating terms
of P(ν ν ) and P(ν ν ) can simply be obtained from Eq. (15) 2:
µ → τ µ → τ
∆
P(ν ν ) sin22θ sin2 23 2 J23 +J13 sin∆ ,
µ → τ ≈ 23 2 − (cid:16) µτ µτ(cid:17) 23
∆
P(ν ν ) sin22θ sin2 23 + 2 J23 +J13 sin∆ , (16)
µ → τ ≈ 23 2 (cid:16) µτ µτ(cid:17) 23
where the good approximation ∆ ∆ has been used in view of the experimental fact
13 ≈ 23
∆m2 ∆m2 ∆m2 [1–4], and the sub-leading and CP-conserving “zero-distance”
| 13| ≈ | 23| ≫ | 12|
1Note that the signs of three CP-violating terms in our Eq. (15) are opposite to those in Eq. (18)
of Ref. [16]. The latter might result from a misprint.
2Note that the CP-violating term shown in our Eq. (16) is apparently different from that given
in Eq. (12) of Ref. [21], where a very different parametrization of the unitarity violation of V has
been adopted.
7
effect [16] has been omitted. Taking account of Eq. (14), we find J23 +J13 = J23 J31
µτ µτ µτ − µτ ≈
ImZ/2. To be more specific,
1
J23 +J13 [s s sin(δ δ δ ) + s s sin(δ δ δ )
µτ µτ ≈ 2 24 34 24 − 34 − 23 25 35 25 − 35 − 23
+ s s sin(δ δ δ )] . (17)
26 36 26 − 36 − 23
In the assumption of s s 0.1 and (δ δ ) 1 (for l = 4,5,6), this non-trivial
2l ∼ 3l ∼ 2l − 3l ∼
CP-violating quantity can reach the percent level.
FIG.1illustratestheCP-violatingasymmetrybetweenν ν andν ν oscillations,
µ → τ µ → τ
where θ π/4, ∆m2 2.5 10−3 eV2 and (J23 +J13) 0.01 have typically been input.
23 ∼ 23 ∼ × µτ µτ ∼
One can see that it is possible to measure this asymmetry in the range L/E (100 400)
< ∼ ···
km/GeV, if the experimental sensitivity is 1%. A short- or medium-baseline neutrino
factory with the beam energy E being abov∼e m 1.78 GeV is expected to have a good
τ ≈
chance to probe or constrain the effect of CP violation induced by the non-unitarity of V
(see Ref. [21] for some detailed discussions).
4 NomatterwhetheraneutrinomassmodelisofType-IseesaworType-IIseesaw, nine
rotation angles and nine phase angles are in general needed to parametrize the correlation
between the charged current interactions of three heavy Majorana neutrinos (N ,N ,N )
1 2 3
and their light counterparts (ν ,ν ,ν ). To reduce the number of free parameters, one
1 2 3
may consider the “unbalanced” seesaw scenarios in which the number of heavy Majorana
neutrinos is smaller than that of ligh Majorana neutrinos [22]. There are at least two
phenomenologically viable scenarios of this nature: one is the minimal Type-I seesaw model
with two heavy Majorana neutrinos [23] and the other is the minimal Type-II seesaw model
with one heavy Majorana neutrino and one Higgs triplet [24]. In either case, we can arrive
at the simplified correlation between V and R.
(1) The minimal Type-I seesaw model with two heavy Majorana neutrinos. In this case,
M is 2 2, M is 3 2, and M = 0 holds in the overall 5 5 neutrino mass matrix .
R × D × L × M
Switching off the rotation matrices O (for i = 1, ,5) in Eq. (8), we are able to fully
i6 ···
parametrize the 3 3 matrix A and the 3 2 matrix R in terms of six rotation angles (θ
× × i4
and θ for i = 1,2,3) and six phase angles (δ and δ for i = 1,2,3):
i5 i4 i5
c c 0 0
14 15
A =  c sˆ sˆ∗ sˆ sˆ∗ c c c 0  ,
− 14 15 25 − 14 24 25 24 25
 c sˆ c sˆ∗ +sˆ sˆ∗ sˆ sˆ∗ sˆ c sˆ∗ c c sˆ sˆ∗ sˆ sˆ∗ c c c 
− 14 15 25 35 14 24 25 35 − 14 24 34 35 − 24 25 35 − 24 34 35 34 35
sˆ∗ c sˆ∗
14 15 15
R =  sˆ∗ sˆ sˆ∗ +c sˆ∗ c c sˆ∗  . (18)
− 14 15 25 14 24 25 15 25
 sˆ∗ sˆ c sˆ∗ c sˆ∗ sˆ sˆ∗ +c c sˆ∗ c c c sˆ∗ 
− 14 15 25 35 − 14 24 25 35 14 24 34 35 15 25 35
It is therefore straightforward to obtain the correlation between V = AV and R.
0
(2) The minimal Type-II seesaw model with one heavy Majorana neutrino and one Higgs
triplet. In this case, M is 1 1, M is 3 1, and M is 3 3 in the overall 4 4 neutrino
R × D × L × ×
mass matrix . Switching off the rotation matrices O (for i < 5) and O (for i < 6) in
M i5 i6
Eq. (8), we can parametrize the 3 3 matrix A and the 3 1 matrix R in terms of three
× ×
rotation angles (θ for i = 1,2,3) and three phase angles (δ for i = 1,2,3) as follows:
i4 i4
8
c 0 0
14
A =  sˆ sˆ∗ c 0  ,
− 14 24 24
 sˆ c sˆ∗ sˆ sˆ∗ c 
− 14 24 34 − 24 34 34
sˆ∗
14
R =  c sˆ∗  . (19)
14 24
c c sˆ∗ 
 14 24 34
Of course, the correlation between V = AV and R is more obvious in this scenario. Taking
0
the Jarlskog invariant J23 for example, we find
eµ
J23 = s c s c2 c [c c c s c sin(δ δ δ )
eµ 12 13 13 14 24 12 13 23 23 24 13 − 12 − 23
c s s c s sin(δ δ δ )+s s s s sin(δ δ +δ δ )] , (20)
− 12 13 14 23 24 14 − 12 − 24 12 14 23 24 14 − 13 23 − 24
in which the first term is essentially governed by the phase parameters of V , but the second
0
and third terms result from the unitarity-violating effects. One may similarly calculate
the other Jarlskog invariants of CP violation. As for ν ν and ν ν oscillations
µ → τ µ → τ
with short or medium baselines, the approximate probabilities obtained in Eq. (16) are
generally applicable and the CP-violating quantity (J23 +J13) may unambiguously signify
µτ µτ
the non-unitarity of V or the existence of certain non-standard neutrino interactions.
5 In summary, we have investigated how the charged current interactions of light
and heavy Majorana neutrinos are correlated with each other in the Type-I and Type-II
seesaw models. It is the first time that an explicit parametrization of this correlation, which
is independent of any details of the seesaw models, has been presented to bridge the gap
between neutrino physics and collider physics. The rotation angles and phase angles in
such a parametrization are expected to be measured or constrained in the precision neutrino
oscillation experiments and by exploring possible signatures of heavy Majorana neutrinos
at the LHC and ILC. We have taken two special but viable examples, the minimal Type-I
seesaw model with two heavy Majorana neutrinos and the minimal Type-II seesaw model
with one heavy Majorana neutrino and one Higgs triplet, to illustrate the simplified V-R
correlation. The implications of R = 0 on the low-energy neutrino phenomenology, such
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as the neutrinoless double-beta decay, the tritium beta decay and CP violation in neutrino
oscillations, have also been discussed. In particular, we have demonstrated that the non-
unitarity of V is possible to give rise to an appreciable CP-violating asymmetry between
ν ν andν ν oscillationswithshortormediumbaselines. Ourgenericresultsremain
µ → τ µ → τ
valid even if the Type-I or Type-II seesaw mechanism is embedded in the supersymmetric
standard model and some other extensions of the standard model.
How to naturally realize an appreciable correlation between V and R in a TeV-scale
seesaw model is actually a real challenge to model builders. The reason is simply that the
main textures of M and M in the Type-I seesaw scenarios or those of M , M and M in
D R D L R
the Type-II seesaw scenarios, which are relevant to possibly observable collider signatures,
aredifficult to imprint onthose sub-leading effects (due to explicit perturbations orradiative
corrections) responsible for the tiny masses of light Majorana neutrinos [12,18]. Much more
efforts are certainly needed to build viable and natural seesaw models at the TeV scale. One
may even speculate the possibility to naturally achieve the TeV-scale leptogenesis and to
experimentally test it at the LHC and ILC.
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Let us stress that testing the unitarity of the light Majorana neutrino mixing matrix in
neutrino oscillations and searching for the signatures of heavy Majorana neutrinos at TeV-
scale colliders can be complementary to each other, both qualitatively and quantitatively, in
order to deeply understand the intrinsic properties of Majorana particles. Any experimental
breakthrough in this aspect will pave the way towards the true theory of neutrino mass
generation and flavor mixing.
The author would like to thank S. Zhou for useful discussions. This work was supported
in part by the National Natural Science Foundation of China.
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