A four-neutrino texture implying bimaximal flavor mixing and reduced LSND effect

8 Pages·0.1 MB·English

by  Wojciech Krolikowski

A four–neutrino texture implying bimaximal flavor mixing
and reduced LSND effect
∗
Wojciech Kro´likowski
Institute of Theoretical Physics, Warsaw University
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Hoz˙a 69, PL–00–681 Warszawa, Poland
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1
Abstract

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v A four–neutrino effective texture is described, where a sterile neutrino mixes nearly
3
2 maximally with the electron neutrino and so, is responsible for the deficit of solar ν ’s
e
0
1 (according to the large–angle MSW solution or vacuum solution, of which the former is
0
0 selected a posteriori). But, while maximal mixing of muon neutrino with tauon neutrino
0
/
h causes the deficit of atmospheric ν ’s, the original magnitude of LSND effect is reduced
µ
p
- by as much as four orders, becoming unobservable.
p
e
h PACS numbers: 12.15.Ff , 14.60.Pq , 12.15.Hh .
:
v
i
X
r
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January 2000
∗
Work supported in part by the Polish KBN–Grant 2 P03B 052 16 (1999–2000).
As is well known, in addition to three active neutrinos ν , ν , ν , one sterile neutrino
e µ τ
ν , at least, is needed to explain in terms of neutrino oscillations three neutrino effects:
s
the deficits of solar ν ’s and atmospheric ν ’s as well as the possible LSND excess of ν ’s
e µ e
in accelerator beam of ν ’s [1]. This is a phenomenological reason for introducing sterile
µ
neutrinos. From the theoretical viewpoint, however, sterile neutrinos may exist in Nature,
whether the LSND effect is real or not.
In this paper, we describe a four–neutrino effective texture implying bimaximal mixing
of ν with ν and ν with ν , but, at the same time, only a tiny LSND effect, reduced by
e s µ τ
as much as four orders of magnitude in comparison with its original estimation.
In our texture, the mass matrix for active neutrinos ν , ν , ν gets the same form
e µ τ
M = (M ) (α, β = e, µ, τ) as the mass matrix for charged leptons e , µ , τ (only the
αβ − − −
values of parameters are expected to be different). In order to operate with an explicit
model, we accept in both cases the ansatz [2]
µε 2α 0
1
(M ) = 2α 4µ(80+ε)/9 8√3α , (1)
αβ  
29
0 8√3α 24µ(624+ε)/25
 
 
where ε > 0, µ > 0 and α > 0 are three parameters, taking different values for neutrinos
and charged leptons.
In the case of charged leptons, the ansatz (1) leads for α 0 to the prediction
→
m 1776.80 MeV, ε 0.172329, µ 85.9924 MeV, (2)
τ
→ → →
if experimental values of m and m are used as an input. In fact, the lowest perturbative
e µ
calculation with respect to α/µ, when applied to the eigenvalue equation for the matrix
(1), gives in particular [2]
2
6 α
m = (351m 136m )+10.2112 MeV
τ µ e
125 − µ!
2
α
= 1776.80+10.2112 MeV . (3)
 µ! 
 
When the experimental value mexp = 1777.05+0.29 [3] is used, Eq. (3) implies
τ 0.26
−
1
2
α
= 0.024+0.028 , (4)
µ! −0.025
what is not inconsistent with α = 0 (then M becomes diagonal). Impressive agreement
of the prediction for m with the experimental mexp is our phenomenological motivation
τ τ
for the use of form (1) as the lepton mass matrix M. Methodologically, we consider here
our form (1) of M as a detailed ansatz, though it can be somehow theoretically supported
(the interested reader may find some arguments in Appendix to Ref. [4]).
In contrast to the charged–lepton case, where α/µ 1 (and so, M is nearly diagonal),
≪
we conjecture in the neutrino case that µ/α 1 (and it is small enough to get M nearly
≪
off–diagonal). The reason is that only in such a situation we can expect nearly maximal
neutrino mixing, namely of ν with ν as it is preferably suggested by Super–Kamiokande
µ τ
experiments on the deficit of atmospheric ν ’s [5]. Then, in order to explain potentially
µ
also the deficit of solar ν ’s [6] as well as the possible LSND effect for accelerator ν ’s [7],
e µ
we accept the popular hypothesis [1] that in Nature there is a sterile neutrino ν which
s
may mix with active neutrinos ν , ν , ν , dominantly with ν .
e µ τ e
To construct an effective model of four–neutrino texture, we assume that the mass
matrix for neutrinos ν , ν , ν , ν has the 4 4 form M = (M ) (α, β = s, e, µ, τ),
s e µ τ αβ
×
where
M = 0 , M = λM = M , M = 0 = M , M = 0 = M (5)
ss se eµ es sµ µs sτ τs
are seven new matrix elements, while the rest of them are old, as given in Eq. (1). Here,
the ratio λ M /M > 0 is a neutrino fourth free parameter. The old neutrino free
se eµ
≡
parameter ε will be put zero (as seen from Eq. (2), even for charged leptons ε is small).
Then,
4 µ 24 µ
M = 0 , M = 80 , M = 624 . (6)
ee µµ ττ
9 29 25 29
The ratios
M µ M 1
ττ µµ
ξ = 299.52 , χ = ξ (7)
≡ M α ≡ M 16.848
eµ eµ
are small, when µ/α 1 is small enough.
≪
2
Now, solving the eigenvalue equation for the 4 4 matrix M in the first perturbative
×
order with respect to ξ, we obtain the following neutrino masses:
2α 1 1/2 1 1 2α 48 1 1
m = 49+λ2 (49 λ2)2 +4λ2 + ξ λ+ ξ ,
0
29 (−√2 − − 249 ) ≃ 29 −s49 249 
(cid:20) q (cid:21)
 
2α 1 1/2 1 1 2α 48 1 1
m = 49+λ2 (49 λ2)2 +4λ2 + ξ λ+ ξ ,
1
29 ( √2 − − 249 ) ≃ 29 s49 249 
(cid:20) q (cid:21)
2α 1 1/2 1 48  
m = 49+λ2+ (49 λ2)2 +4λ2 + ξ +χ
2
29 (−√2 (cid:20) q − (cid:21) 2 (cid:18)49 (cid:19))
2α 1 48
7+ ξ +χ ,
≃ 29 − 2 49
(cid:20) (cid:18) (cid:19)(cid:21)
2α 1 1/2 1 48
m = 49+λ2 + (49 λ2)2 +4λ2 + ξ +χ
3
29 ( √2 (cid:20) q − (cid:21) 2 (cid:18)49 (cid:19))
2α 1 48
7+ ξ +χ . (8)
≃ 29 2 49
(cid:20) (cid:18) (cid:19)(cid:21)
Here, the second step is valid in the linear approximation in λ, what requires small λ/49,
while the former perturbative calculation with respect to ξ works for small ξ/7. We can
> >
conclude from Eqs. (8) that m m m m .
3 2 1 0
∼ | | ≫ ∼ | |
The neutrino diagonalizing 4 4 matrix U = (U ) (α = s, e, µ, τ, i = 0, 1, 2, 3),
αi
×
such that U MU = diag (m , m , m , m ), gets in the zero order with respect to ξ and
† 0 1 2 3
in the linear approximation in λ the following form:
1 1 λ λ
√2 √2 49√2 49√2
√48 √48 1 1
 
(U ) −7√2 7√2 −7√2 7√2 +O(ξ/7) . (9)
αi ≃  λ λ 1 1 
 −49√2 −49√2 √2 √2 
 1 1 √48 √48 
 
 7√2 −7√2 −7√2 7√2 
 
Evidently, in this case ξ/7 ought to be smaller than λ/49. If the charged–lepton diagonal-
izing 3 3 matrix is nearly unit due to the small value (4) of α/µ, the lepton counterpart
×
V = (V ) of the quark Cabibbo—Kobayashi—Maskawa matrix is approximately equal
iα
to U = U = (U ). Thus, in this approximation, the fields
† †iα α∗i
(cid:16) (cid:17)
ν = V ν = U ν (10)
i iα α α∗i α
α α
X X
describe four massive neutrinos ν (i = 0,1,2,3) in terms of four flavor neutrinos ν (α =
i α
s, e, µ, τ). Hence,
3
ν = U ν , ν = ν 0 = U ν . (11)
α αi i | αi α†| i α∗i| ii
i i
X X
Then, the neutrino oscillation probabilities on the energy shell E read
P (ν ν ) = ν eiPL ν 2
α β β α
→ |h | | i|
= δ 4 U U U U sin2x , (12)
βα − β∗j αj βi α∗i ji
j>i
X
where L denotes the experimental baseline and
∆m2L
x = 1.27 ji , ∆m2 = m2 m2 (13)
ji E ji j − i
with ∆m , L and E expressed in eV, km and GeV, respectively. In Eq. (12) the eigen-
ji
values of momentum operator P are p = E2 m2 E m2/2E. Evidently, because
i − i ≃ − i
q
of real M and thus real U , the possible CP violation is here neglected.
αβ αi
From Eqs. (12) and (9) we calculate in the zero perturbative order with respect to ξ
and linear approximation in λ the following oscillation probabilities:
482 4 48 1
P (ν ν ) 1 sin2x · sin2x sin2x ,
e → e ≃ − 492 10 − 492 21 − 492 32
P (ν ν ) 1 sin2x ,
µ µ 32
→ ≃ −
1
P (ν ν ) sin2x . (14)
µ e 32
→ ≃ 49
In the first and third formula (14) we put approximately ∆m2 ∆m2 ∆m2 ∆m2
20 ≃ 30 ≃ 21 ≃ 31
due to Eqs. (8) with ξ 0 (then, a linear term in λ appearing in the third formula
≃
vanishes). Note from Eqs. (8) that
2 48 2α 2 2α 2 48
∆m2 λξ , ∆m2 14 ξ +χ (15)
10 ≃ 49s49 29 32 ≃ 49 49
(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)
for small λ/49 and ξ/7. Here, χ = 5.9354 10 2ξ.
−
×
If 1.27∆m2 L /E = O(1) and ∆m2 ∆m2 3 10 3 eV2 [5], the second
32 atm atm 32 ↔ atm ∼ × −
formula (14) is able to describe oscillations of atmospheric ν ’s (dominantly into ν ’s)
µ τ
with maximal amplitude. Then, the second Eq. (15) gives the estimate
4
α2ξ 4.3 10 2 eV2 . (16)
−
∼ ×
Hence, if one assumes reasonably that α O(1 eV), one gets ξ O(10 2).
−
≤ ≥
On the other hand, if 1.27∆m2 L /E = O(1) and ∆m2 ∆m2 10 5 eV2 or
10 sol sol 10 sol −
↔ ∼
10 10 eV2 [6],thefirstformula(14)candescriberespectively large–angleMSWoscillations
−
or vacuum oscillations of solar ν ’s (dominantly into ν ’s) with nearly maximal amplitude.
e s
In fact, it implies
482 4 48+1 482
P (ν ν ) 1 sin2x · 1 sin2x (17)
e → e ≃ − 492 10 − 2 492 ≃ − 492 10
·
due to x x x . Then, from the first Eq. (15) we get the estimate
10 32 21
≪ ≪
α2λξ 5.2 10 2 eV2 or 5.2 10 7 eV2 , (18)
− −
∼ × ×
respectively.
Thus, we find from Eqs. (16) and (18) that
λ 1.2 or 1.2 10 5 , (19)
−
∼ ×
respectively. This shows that the matrix element M is comparable or small versus M .
se eµ
Evidently, only the first option (related to large–angle MSW oscillations of solar ν ’s)
e
can be compatible with the mixing matrix (9) and so, with the oscillation formulae (14)
leading to the nearly maximal mixing of ν with ν . In fact, only in this option, ξ may be
s e
smaller than λ, as required by the form (9) of neutrino mixing matrix.
In the case of Chooz experiment searching for oscillations of reactor ν¯ ’s [8], where it
e
happens that 1.27∆m2 L /E = O(1), the first formula (14) leads to
32 Chooz Chooz
1 2 48
P (ν¯ ν¯ ) 1 sin2x · 1 (20)
e → e ≃ − 492 32 − 492 ≃
since x x x . This is consistent with the negative result of Chooz experiment.
10 32 21
≪ ≪
The third formula (14) implies te existence of ν ν neutrino oscillations with the
µ e
→
amplitude equal to 1/49 0.02 and the mass–squared scale given by ∆m2 . Such an
≃ 32
amplitude is compatible with the LSND estimation, say, sin22θ 0.02, but the
LSND
∼
5
mass–squared scale ∆m2 — being equal to the atmospheric ∆m2 3 10 3 eV2 — is
32 atm ∼ × −
smaller than the LSND estimation, say, ∆m2 0.5 eV2 [7] roughly by two orders of
LSND ∼
magnitude.
In conclusion, our four–neutrino effective texture may describe correctly both deficits
of solar ν ’s and atmospheric ν ’s. Then, it predicts the existence of a tiny LSND effect of
e µ
the magnitude reduced by four orders in comparison with the original LSND estimation.
It is so, because
∆m2 L 10 2∆m2 L
sin2 1.27 32 LSND sin2 1.27 − LSND LSND 10 4 (21)
−
ELSND ! ∼ ELSND ! ∼
for 1.27∆m2 L /E 1. This reduced LSND effect would be, therefore, practi-
32 LSND LSND ∼
cally unobservable (for original L = L and E = E ).
LSND LSND
Obviously, the experimental problem of existence of the LSND effect, or of another
realization of ν ν neutrino oscillations, is crucial for all discussions about neutrino
µ e
→
texture. In particular, a clear confirmation of the original LSND effect would exclude our
four–neutrino effective texture.
In such a case, the option of three pseudo–Dirac neutrinos might be invoked to explain
all three neutrino–oscillation effects: the deficits of solar ν ’s and atmospheric ν ’s as
e µ
well as the LSND effect (cf. e.g., Refs. [4] and [9]). This option involves three natural
MajoranasterileneutrinosmixingnearlymaximallywiththreeMajoranaactiveneutrinos,
andproducesthreepairsoflight mass–neutrino states. Itisincontrasttothepopularsee–
saw option, where the natural Majorana sterile neutrinos and Majorana active neutrinos
practically do not mix, and where they produce heavy and light mass–neutrino states,
respectively. In the see–saw option, small masses of the latter states are conditioned by
large masses of the former.
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3. Review of Particle Physics, Eur. Phys. J. C 3, 1 (1998).
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(to appear); and references therein.
5. Y. Fukuda et al. (Super–Kamiokande Collaboration), Phys. Rev. Lett. 81, 1562 (1998).
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therein.
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