Yukawaon model and unified description of quark and lepton mass matrices

0.07 MB·English

by  Yoshio Koide

OU-HET-652/2010
Yukawaon model and unified description of
quark and lepton mass matrices1
Yoshio Koide
Department of Physics, Osaka University, Toyonaka, Osaka 560-0043, Japan
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E-mail address: [email protected]
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2 Abstract

n
In the so-called yukawaon model, where effective Yukawa coupling constants Yeff (f =
a f
J e,ν,u,d) are given by vacuum expectation values of gauge singlet scalars (yukawaons) Y
f
3
with 3 3 flavor components, it is tried to give a unified description of quark and lepton
1 ×
mass matrices. Especially, without assuming any discrete symmetry in the lepton sector,

] nearly tribimaximal mixing is derived by assumed a simple up-quark mass matrix form.
h
p
-
p
1 What is a yukawaon model?
e
h
[
First, let us give a short review of the so-called yukawaon model: We regard Yukawa coupling

1 eff
constants Y as effective coupling constants Y in an effective theory, and we consider that
v f f
5 Yeff originate in vacuum expectation values (VEVs) of new gauge singlet scalars Y , i.e.
f f
7
0
2
y
. Yeff = f Y , (1)
1 f Λh fi
0
0 where Λ is a scale of an effective theory which is valid at µ Λ, and we assume Y Λ.
f
1 ≤ h i ∼
: We refer the fields Y as yukawaons [1] hereafter. Note that the effective coupling constants
v f
eff
i Y evolve as those in the standard SUSY model below the scale Λ, since a flavor symmetry is
X f
completely broken at a high energy scale µ Λ.
r
a ∼
In the present work, we assume an O(3) flavor symmetry. In order to distinguish each Y
f
from others, we assume a U(1) symmetry (i.e. sector charge). (The SU(2) doublet fields q, ℓ,
X L
H and H are assigned to sector charges Q = 0.) Then, we obtain VEV relations as follows:
u d X
(i) We give an O(3) and U(1) invariant superpotential for yukawaons Y . (ii) We solve SUSY
X f
vacuum conditions ∂W/∂Y = 0. (iii) Then, we obtain VEV relations among Y .
f f
For example, in the seesaw-type neutrino mass matrix, M Y Y −1 Y T, we obtain
ν ν R ν
∝ h ih i h i
[2]
Y Y Φ + Φ Y (2)
R e u u e
h i ∝ h ih i h ih i
together with Y Y and Y Φ Φ , i.e. a neutrino mass matrix is given by
ν e u u u
h i ∝ h i h i∝ h ih i
−1
M Y Y Φ + Φ Y Y , (3)
ν e e e e e u e u e e e e e
h i ∝ h i {h i h i h i h i } h i
where Φ diag(√m ,√m ,√m ), and A denotes a form of a VEV matrix A in the
u u u c t f
h i ∝ h i h i
diagonal basis of Y (we refer it as f basis). We can obtain a form Φ = V(δ)T Φ V(δ)
f u d u u
h i h i h i
1
To appear in theProceedings of Lepton-Photon 2009.
from the definition of the CKM matrix V(δ), but we do not know an explicit form of Φ .
u e
h i
Therefore, in a previous work [2], the author put an ansatz, Φ = V(π)T Φ V(π) by
u e u u
h i h i
supposing Φ Φ , and he obtained excellent predictions of the neutrino oscillation
u e u d
h i ≃ h i
parameters without assuming any discrete symmetry. However, there is no theoretical ground
for the ansatz for the form Φ .
u e
h i
The purpose of the present work is to investigate a quark mass matrix model in order to
predict neutrino mixing parameters on the basis of a yukawaon model (2), without such the ad
hoc ansatz, because if we give a quark mass matrix model where mass matrices (M ,M ) are
u d
given on the e basis, then, we can obtain the form Φ by using a transformation Φ =
u e u e
h i h i
U Φ UT, where U is defined by UTM U = Mdiag.
uh uiu u u u u u u
2 Yukawaons in the quark sector
We assume a superpotential in the quark sector [3]:
ξ
W = µ [Y Θ ]+λ [Φ Φ Θ ]+µX[Φ ΘX]+µX[Y ΘX]+ q[Φ (Φ +a E)Φ ΘX]. (4)
q u u u u u u u u u u d d d X Λ e X q e q
q=u,d
Here and hereafter, for convenience, we denotes Tr[...] as [....] simply. From SUSY vacuum
conditions ∂W/∂Θ =0, ∂W/∂ΘX = 0 and ∂W/∂ΘX = 0, we obtain Y Φ Φ ,
u u d h ui ∝ h uih ui
M1/2 Φ Φ ( Φ +a E ) Φ , (5)
u ∝ h uie ∝ h eie h Xie uh ie h eie
M Y Φ ( Φ +a E ) Φ , (6)
d d e e e X e d e e e
∝ h i ∝ h i h i h i h i
respectively. Here, Φ and E are given by
X e e
h i h i
1 1 1 1 0 0
1   
Φ X 1 1 1 , E 1 0 1 0 . (7)
X e e
h i ∝ ≡ 3  h i ∝ ≡  
 1 1 1   0 0 1 
(Note thattheVEVform ΦX e breaks theO(3) flavor symmetry into S3.) Therefore, weobtain
h i
quark mass matrices
M1/2 M1/2(X +a 1)M1/2, M M1/2 X +a eiαd1 M1/2, (8)
u ∝ e u e d ∝ e (cid:0) d (cid:1) e
on the e basis. Note that we have assumed that the O(3) relations are valid only on the e and
u bases, so that Y and Y must be real.
e u
h i h i
A case au 0.56 can give a reasonable up-quark mass ratios mu1/mu2 = 0.043 and
≃ − p
pmu2/mu3 = 0.057, which are in favor of the observed values [4] pmu/mc = 0.045+−00..001130, and
m /m = 0.060 0.005 at µ =M .
c t Z
p ±
Sector Parameters Predictions
sin2θatm tan2θsolar U13
| |
M ξ = +0.0005 0.982 0.449 0.012
ν
ξ = 0.0012 0.990 0.441 0.014
−
M1/2 a = 0.56 mu = 0.0425 mc = 0.0570
u u − qmc qmt
two parameters 5 observables: fitted excellently
M a eiαd md, ms, V , V , V , V
d d qms qmb | us| | cb| | ub| | td|
two parameters 6 observables: not always excellent
Table 1: Summary of the present model.
3 Yukawaons in the neutrino
However, the up-quark mass matrix (5) failed to give reasonable neutrino oscillation parameter
values although it can give reasonable up-quark mass ratios. Therefore, we will slightly modify
the model (2) in the neutrino sector.
1/2
Note that the sign of the eigenvalues of M given by Eq.(8) is (+, ,+) for the case
u
−
a 0.56. If we assume that the eigenvalues of Φ must be positive, so that Φ in
u u u u u
≃ − h i h i
Eq.(2) is replaced as Φ Φ diag(+1, 1,+1), then, we can obtain successful results
u u u u
h i → h i · −
except for tan2θsolar, i.e. predictions sin22θatm = 0.984 and U13 = 0.0128 and an unfavorable
| |
prediction tan2θ = 0.7033.
solar
When we introducea new field P with aVEV P diag(+1, 1,+1), we mustconsider
u u u
h i ∝ −
an existence of P Y Φ +Φ Y P in addition to Y P Φ +Φ P Y , because they have the same
u e u u e u e u u u u e
U(1) charges. Therefore, we modify Eq.(2) into
X
λ
R
W = µ [Y Θ ]+ [(Y P Φ +Φ P Y )Θ ]+ξ[(P Y Φ +Φ Y P )Θ ] , (9)
R R R R e u u u u e R u e u u e u R
Λ { }
which leads to VEV relation Y Y P Φ +Φ P Y +ξ(P Y Φ +Φ Y P ). The results at
R e u u u u e u e u u e u
∝
a 0.56 are excellently in favor of the observed neutrino oscillation parameters by taking a
u
≃ −
small value of ξ (see Table 1):
| |
Also, we can calculate the down-quark sector. We have two parameters (a ,α ) in the
d d
down-quark sector given in Eq.(8). (See Table 2 in Ref.[3]). The results are roughly reasonable,
although Vi3 and V3i are somewhat larger than the observed values. Those discrepancies will
| | | |
be improved in future version of the model.
4 Summary
In conclusion, for the purpose of deriving the observed nearly tribimaximal neutrino mixing, a
possible yukawaon model in the quark sector is investigated. Five observable quantities (2 up-
quark mass ratios and 3 neutrino mixing parameters) are excellently fitted by two parameters.
Also, the CKM mixing parameters and down-quark mass ratios are given under additional 2
parameters. The results are summarized in Table 1.
It is worthwhile to notice that the observed tribimaximal mixing in the neutrino sector is
substantially obtained from the up-quark mass matrix structure (8). Although the model for
down-quark sector still need an improvement, the present approach will provide a new view to
a unified description of the masses and mixings.
References
[1] Y. Koide, Phys.Rev. D78093006 (2008); Phys.Rev.D79 033009 (2009).
[2] Y. Koide, Phys.Lett. B665 227 (2008).
[3] Y. Koide, Phys.Lett. B680 76 (2009).
[4] Z.-z.Xing,H.ZhangandS.Zhou,Phys.Rev.D77(2008) 113016.Also,seeH.FusaokaandY.Koide,Phys.
Rev.D57 (1998) 3986.