Guesswork for Dirac and Majorana neutrino mass matrices

0.14 MB·English

by  W. Krolikowski

IFT(cid:21) 04/3
Guesswork for Dira
and Majorana neutrino
∗
mass matri
es
Woj
ie
h Królikowski
4
0
0
Institute of Theoreti
al Physi
s, Warsaw University
2

n Ho»a 69, PL(cid:21)00(cid:21)681 Warszawa, Poland
a
J

4 Abstra
t
1

1 In the framework of seesaw me
hanism with three neutrino (cid:29)avors, we propose ten-
v
1 tatively an e(cid:30)
ient parametrization for the spe
tra of Dira
and righthanded Majorana
0
1 neutrino mass matri
es in terms of three free parameters. Two of them are related to
1
0 (and determined by) the
orresponding parameters introdu
ed previously for the mass
4
0
spe
tra of
harged leptons and up and down quarks. The third is determined from the
/
h ∆m2 ∆m2
p experimentalestimate of solar 21. Then, the atmospheri
32 ispredi
ted
loseto its
-
p
experimental estimation. With the use of these three parameters all light a
tive-neutrino
e
h m < m < m M < M < M
1 2 3 1 2 3
: masses and heavy sterile-neutrino masses are readily
v
i
X evaluated. The latter turn out mu
h more hierar
hi
al than the former. The lightest
M O(106 GeV)
r 1
a heavy mass
omes out to be of the order so, it is too light to imply that
the me
hanism of baryogenesis through thermal leptogenesis might work.
PACS numbers: 12.15.Ff , 12.15.Hh , 14.60.Pq .
January 2004
∗
Work supported in partby the PolishState CommitteeforS
ienti(cid:28)
Resear
h(KBN), grant2P03B
129 24 (2003(cid:21)2004).
e = e ,µ ,τ
i − − −
Some time ago we proposed for
harged leptons an e(cid:30)
ient empiri
al
mass formula [1℄
ε(e) 1
m = ρ µ(e) N2 + − ,
ei i (cid:18) i N2 (cid:19) (1)
i
where
N = 1,3,5 ,
i
(2)
and
1 4 24
ρ = , ,
i
29 29 29 (3)
ρ = 1 µ(e) > 0 ε(e) > 0
( i i ), while and are
onstants. In fa
t, with the experimental
P
m = 0.510999 m = 105.658
e µ
values MeV and MeV as an input, the formula (1) rewritten
expli
itly as
µ(e) µ(e)4 µ(e)24
m = ε(e) , m = (80+ε(e)) , m = (624+ε(e))
e µ τ
29 29 9 29 25 (4)
leads to the predi
tion
6
m = (351m 136m ) = 1776.80 MeV
τ µ e
125 − (5)
and also determines both
onstants
29(9m 4m ) 320m
µ(e) = µ − e = 85.9924 MeV , ε(e) = e = 0.172329.
320 9m 4m (6)
µ e
−
mexp = 1776.99+0.29
τ 0.26
The predi
tion (5) lies really
lose to the experimental value − MeV
[2℄. Though the formula (1) has essentially the empiri
al
hara
ter, there is a spe
ulative
ba
kground for it based on a Kähler-like extension of Dira
equation whi
h the interested
N ρ (i = 1,2,3)
i i
reader may (cid:28)nd in Ref. [1℄. In parti
ular, the numbers and given in
Eqs. (2) and (3) are interpreted there.
The
harged-lepton mass formula [1℄ was re
ently extended to up and down quarks,
u = u,c,t d = d,s,b
i i
and , by introdu
ing an additional term for the third quark genera-
tion, leading to [3℄
1
ε(u) 1
m = ρ µ(u) N2 + − +δ β(u)
ui i (cid:18) i N2 i3 (cid:19) (7)
i
and
ε(d) 1
m = ρ µ(d) N2 + − +δ β(d) ,
di i (cid:18) i N2 i3 (cid:19) (8)
i
N ρ µ(u,d) > 0 ε(u,d) > 0
i i
where and are given as before in Eqs. (2) and (3), while , and
β(u,d) > 0
are
onstants. It is seen that a priori Eqs. (7) and (8)
annot give us any mass
predi
tions, sin
e there are six quark masses and six free parameters. However, the latter
are uniquely determined. In fa
t, assuming for quark masses their mean experimental
estimates [2℄
m 3 MeV , m 1.2 GeV , m 174 GeV
u c t
∼ ∼ ∼ (9)
and
m 6.75 MeV , m 118 MeV , m 4.25 GeV ,
d s b
∼ ∼ ∼ (10)
one
an
al
ulate
6 24 20 + 0.81 β(u)
m = (351m 136m )+ µ(u,d)β(u,d) GeV
t,b 125 c,s − u,d 29 ∼ (cid:26) 1.94+0.078β(d) (cid:27) (11)
and
29(9m 4m ) 978 320m 0.0890
µ(u,d) = c,s − u,d MeV, ε(u,d) = u,d .
320 ∼ (cid:26) 93.8 (cid:27) 9m 4m ∼ (cid:26) 2.09 (cid:27)
c,s u,d
−
(12)
From Eqs. (11) it follows that
β(u) 190 , β(d) 30 ,
∼ ∼ (13)
thus
β(u)
6.3 .
β(d) ∼ (14)
2
It may be interesting to note that the experimental ratio (14) is ni
ely reprodu
ed by the
ansatz
25/9
β(u,d) (3B +Q(u,d))2 = ,
∝ (cid:26) 4/9 (cid:27) (15)
2/3
B = 1/3 Q(u,d) =
where and (cid:26) 1/3 (cid:27) are the baryon number and ele
tri

harge of
−
β(u)/β(d)= 6.25
quarks. Then, . Note also that the analogi
al
onstant for
harged leptons
β(e) (L + Q(e))2 = 0 L = 1 Q(e) = 1
vanishes, ∝ , where and − are their lepton number
F = 3B+L
and ele
tri

harge ( is the fermion number as de(cid:28)ned for quarks and
harged
leptons).
In thepresent notewe dis
uss themassspe
trum ofthreea
tive (lefthanded)neutrinos
ν = ν ,ν ,ν ν = ν ,ν ,ν
αL eL µL τL iL 1L 2L 3L
related to their mass states through the unitary
mixing transformation
ν = U ν ,
αL αi iL
(16)
Xi
U = (U )
αi
where the neutrino mixing matrix is experimentally
onsistent with the bilarge
form [4℄
c s 0
12 12
U =  −√12s12 √12c12 √12  , (17)
1 s 1 c 1
 √2 12 −√2 12 √2 
c = cosθ s = sinθ θ 33 c = cosθ = 1/√2
12 12 12 12 12 ◦ 23 23
where and with large ∼ , while
s = sinθ = 1/√2 θ = 45
23 23 23 ◦
and with maximal . In Eq. (17) the matrix element
U = s exp( iδ) s = sinθ s2 < 0.03
e3 13 − is negle
ted, where 13 13 with 13 . Three sterile
ν = ν ,ν ,ν ν = ν ,ν ,ν
αR eR µR τR iR 1R 2R 3R
(righthanded) neutrinos and their mass states
appear as a ba
kground.
6 6
Our starting point will be the generi
× mass matrix
0 M(D)
(cid:18) M(D)T M(R) (cid:19) (18)
ν ν
αL αR
(in the basis of a
tive and sterile ), involving Dira
and righthanded Majorana
3 3 M(D) M(R)
× mass matri
es, and . A

epting the familiar seesaw me
hanism [5℄ we
3 3
will use for a
tive neutrinos the e(cid:27)e
tive Majorana × mass matrix of the form
3
M(ν) = M(D)M(R)−1M(D)T ,
(19)
M(R) M(D) m(D) = m(D),
νi ν1
where is assumed to dominate over . For the eigenvalues
m(D),m(D) M(D)
ν2 ν3
of the Dira
neutrino mass matrix we will a

ept tentatively the for-
mula of the same type as Eq. (1) for
harged leptons,
ε(ν) 1
m(D) = ρ µ(ν) N2 + − ,
νi i (cid:18) i N2 (cid:19) (20)
i
µ(ν) > 0 ε(ν) > 0
where and are new
onstants.
M(e)
In the (cid:29)avor representation, where the
harged-lepton mass matrix is diagonal,
U
the neutrino mixing matrix is at the same time the neutrino diagonalizing matrix,
U M(ν)U = diag(m ,m ,m ).
† ν1 ν2 ν3 (21)
M(ν) = M(ν) U = U
∗ ∗
Here, for simpli
ity, and [as in Eq. (17)℄. In the
ase of seesaw
M(ν) U
form (19) of , it seems natural to
onje
ture that is also the diagonalizing matrix
M(D)
for the Dira
neutrino mass matrix ,
U M(D)U = diag(m(D),m(D),m(D)).
† ν1 ν2 ν3 (22)
M(R)−1 U
Noti
e that then the inverse in Eq. (19) is diagonalized by as well, sin
e from
M(R)−1 = M(D)−1M(ν)M(D)T−1 M(D)−1 m(νDi ) = 0
Eq. (19) (if the inverse exists i.e., all 6 )
M(ν) M(D)−1 U
and both and on the rhs are diagonalized by . Hen
e
U M(R)U = diag(M ,M ,M ),
† ν1 ν2 ν3 (23)
when Eq. (22) is
onje
tured.
M(ν)
Thus, under the
onje
ture (22) we obtain from the seesaw form (19) of the
ν
iL
following Majorana mass spe
trum for light a
tive (lefthanded) neutrinos :
(D)2
m
m = νi
νi M (24)
νi
(D)
M m m M
with νi ≫ νi ≫ νi,where νi aretheMajoranamassesofheavysterile(righthanded)
ν M(R) = M(R)
iR ∗
neutrinos . Here, for simpli
ity, .
4
(D)
m m
In order to pro
eed further with
al
ulations of νi [from Eq. (20)℄ and νi [from
µ(ν) ε(ν)
Eq. (24)℄ we are for
ed to make some tentative
onje
tures about and as well as
M
νi. We will propose tentatively that
µ(ν) : µ(e) = µ(u) : µ(d) , ε(ν) : ε(e) = ε(u) : ε(d)
(25)
and also
M N2m(D),
νi ∝ i νi (26)
N = 1,3,5
i
where as before in Eq. (2) [in Ref. [6℄ we
onje
tured tentatively that
M N2m
νi ∝ i ei instead of Eqs (26)℄. With the use of values (6) and (12), Eqs. (25) imply
that
µ(ν) 896 MeV , ε(ν) 7.35 10 3.
−
∼ ∼ × (27)
Then, the mass formula (20) gives the following hierar
hi
al estimates of Dira
neutrino
masses:
µ(ν) µ(ν)
m(D) = ε(ν) 0.227 MeV m ,
ν1 29 ∼ ≪ µ(e) e
µ(ν)4 µ(ν)
m(D) = 80+ε(ν) 1.10 GeV m ,
ν2 29 9 ∼ ∼ µ(e) µ
(cid:0) (cid:1)
µ(ν)24 µ(ν)
m(D) = 624+ε(ν) 18.5 GeV m .
ν3 29 25 ∼ ∼ µ(e) τ (28)
(cid:0) (cid:1)
The (weighted) proportionality relation (26), when applied to the seesaw spe
trum (24),
leads to
1
m m(D).
νi ∝ N2 νi (29)
i
(D)
m m
This shows that νi are less hierar
hi
al than νi . From Eqs. (29) we
an see that
(D) (D)
m m m 25m
ν1 = 9 ν1 1.86 10 3 , ν2 = ν2 0.165.
−
mν2 m(νD2) ∼ × mν3 9 m(νD3) ∼ (30)
mexp= (∆m2 )exp √7 10 5 = 8.4 10 3
Thus, takingtheexperimentalestimate ν2 21 ∼ × − eV × − eV
p
as an input, we predi
t from Eqs. (30) that
5
m 1.6 10 5 eV , m √2.6 10 3 eV = 5.1 10 2 eV .
ν1 ∼ × − ν3 ∼ × − × − (31)
m √2.6 10 3 ∆m2 = m2 (mexp)2 2.5 10 3 eV2
The predi
tion ν3 ∼ × − eV gives 32 ν3− ν2 ∼ × −
lose
(∆m2 )exp 2.5 10 3 eV2 (∆m2 )exp
tothe experimentalestimate 32 ∼ × − (the lowerestimate 32 ∼
2 10 3 eV2 m √2.1 10 3
× − would
orrespond to the lower predi
tion ν3 ∼ × − eV).
ζ
Denoting the proportionality
oe(cid:30)
ient in Eq. (26) by , we have
M = ζN2m(D)
νi i νi (32)
m = (1/ζ)m(D)/N2 = (1/ζ2)M /N4 ζ
and νi νi i νi i due to Eq. (29). Thus, may be
al
ulated
e.g. from the relation
ζ = m(D)/9m 1.46 1010,
ν2 ν2 ∼ × (33)
mexp √7 10 5
where the experimental estimate ν2 ∼ × − eV is applied. Then, using the values
(28), we obtain from Eqs. (32) the following hierar
hi
al estimates of Majorana sterile-
neutrino masses:
M = ζm(D) 3.3 106 GeV,
ν1 ν1 ∼ ×
M = 9ζm(D) 1.4 1011 GeV,
ν2 ν2 ∼ ×
M = 25ζm(D) 6.8 1012 GeV.
ν3 ν3 ∼ × (34)
(D) (D)
m M m m
It is seen that νi are less hierar
hi
al than νi (and νi less than νi ).
M ν
Note that the Majorana mass ν1 of the lightest heavy sterile neutrino 1R is too
M > 108
light by two orders of magnitude to rea
h the estimated lower bound ν1 ∼ GeV
required for the working of baryogenesis through thermal leptogenesis [7℄ (of
ourse, in
M(ν) = M(ν) M(R) = M(R)
∗ ∗
this me
hanism 6 and 6 ).
In
on
lusion, our tentative proposal presented here for the Dira
and Majorana
M(D) M(R)
neutrino mass matri
es and
ontains two items: (i) the parametrization
m(D) µ(ν) ε(ν)
νi
(20) of Dira
neutrino masses in terms of two
onstants and determined
through the
onditions (25), and (ii) the parametrization (32) of Majorana neutrino
M ζ
masses νi by one
onstant determined from the experimental estimation of solar
6
∆m2 ∆m2
21 32
. Then, the atmospheri
is predi
ted
lose to its experimental estimate.
m
All neutrino seesaw masses νi and Majorana masses are evaluated. The mass spe
-
(D) (D) (D)
m < m < m m < m < m M < M < M
tra ν1 ν2 ν3, ν1 ν2 ν3 and ν1 ν2 ν3 are hierar
hi
al,
1 : 5.4 102 : 3.3 103 1 : 4.8 103 : 8.2 104 1 : 4.4 104 : 2.0 106
behaving as × × , × × and × × ,
m 1.6 10 5 m(D) 0.23 M 3.3 106
respe
tively, with ν1 ∼ × − eV, ν1 ∼ MeV, and ν1 ∼ × GeV.
7
Referen
es
[1℄ W. Królikowski, A
ta Phys. Pol. B 33, 2559 (2002); and referen
es therein.
[2℄ The Parti
le Data Group, Phys. Rev D 66, 010001 (2002).
[3℄ W. Królikowski, "A proposal of quark mass formula and lepton spe
trum", to appear
in A
ta Phys. Pol. B, (2004).
[4℄ Cf. e.g. V. Barger, D. Marfatia and K. Whisnant, hep(cid:21)ph/0308123; and referen
es
therein.
[5℄ M. Gell-Mann, P. Ramond and R. Slansky, in Supergravity, edited by F. van Nieuwen-
huizen and D. Freedman, North Holland,1979; T. Yanagida, Pro
. of the Workshop on
Uni(cid:28)ed Theory and the Baryon Number in the Universe, KEK, Japan, 1979; R.N. Mo-
hapatra and G. Senjanovi¢, Phys. Rev. Lett. 44, 912 (1980).
[6℄ W. Królikowski, hep-ph/0309020.
[7℄ Cf. in parti
ular G.-F. Giudi
e, A. Notari, M. Raidal, A. Riotto and A. Strumia
hep(cid:21)ph/0302092;T.Hambye,Y.Lin,A.Notari,M.Papu

iandA.Strumiahep(cid:21)ph/0312203;
and referen
es therein.
8