Large $θ_{13}^ν$ and Unified Description of Quark and Lepton Mixing Matrices

0.22 MB·English

by  Yoshio Koide

Large θν and Unified Description
13
of Quark and Lepton Mixing Matrices
Yoshio Koidea and Hiroyuki Nishiurab
a Department of Physics, Osaka University, Toyonaka, Osaka 560-0043, Japan
E-mail address: [email protected]
b Faculty of Information Science and Technology, Osaka Institute of Technology, Hirakata,
2
1 Osaka 573-0196, Japan
0
E-mail address: [email protected]
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t
c
O
Abstract
7
2
We present a revised version of the so-called “yukawaon model”, which was proposed

] for the purpose of a unified description of the lepton mixing matrix UPMNS and the quark
h
p mixingmatrixVCKM. Itisassumedfromaphenomenologicalpointofviewthattheneutrino
- DiracmassmatrixMD isgivenwithasomewhatdifferentstructurefromthe chargedlepton
p
e mass matrix Me, although MD = Me was assumed in the previous model. As a result, the
h 2
revisedmodelpredicts a reasonablevalue sin 2θ13 ∼0.07with keepingsuccessfulresults for
[
other parameters in UPMNS as well as VCKM and quark and lepton mass ratios.
2
v
PCAC numbers: 11.30.Hv, 12.15.Ff, 14.60.Pq, 12.60.-i,
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2 1 Introduction
1
In a series of papers [1, 2, 3], the authors have investigated a unified description of the
.
9
0 lepton mixing matrix [4] UPMNS and the quark mixing matrix [5] VCKM. The essential idea
2
is as follows: (i) The Yukawa coupling constants Y (f = u,d,e, and so on) in the standard
1 f
: model are effectively given by vacuum expectation values (VEVs) of scalars (“yukawaon”) Y
v f
i with 3 × 3 components, i.e. by hY i/Λ. Here Λ is an energy scale of the effective theory.
X f
r (The yukawaon model is a kind of the “flavon” model [6].) (ii) The model does not contain
a
any coefficients which are dependent on the family numbers. The hierarchical structures of
the effective Yukawa coupling constants originate only in a fundamental VEV matrix hΦ0i,
whose hierarchical structure is ad hoc assumed and whose VEV values are fixed by the observed
charged lepton masses. (iii) Relations among those VEV matrices are obtained from SUSY
vacuum conditions for a given superpotential under family symmetries and R charges assumed.
(Since we use the observed charged lepton mass values as the input values, it is a characteristic
in the yukawaon model that adjustable parameters are quite few.)
Inthepreviousmodel[1,3],thequarkandleptonmassmatrices(chargedleptonmassmatrix
M , Dirac neutrino mass matrix M , down-quark mass matrix M , neutrino mass matrix M ,
e D d ν
1
and right-handed Majorana neutrino mass matrix M ) are given as follows:
R
Me = keΦ0(1+aeX3)Φ0,
M = M ,
D e
Md = kd Φ0(1+adX3)Φ0+m0d1 ,
M = k′M(cid:2)ˆ Mˆ , (cid:3) (1.1)
u u u u
Mˆu = kuΦ0(1+auX3)Φ0,
M = M M−1MT,
ν D R D
M = k (Mˆ M +M Mˆ )+··· ,
R R u e e u
where Me, Φ0, X3, ··· are 3×3 numerical matrices which results from VEV matrices of scalar
fields. Here the VEV matrices Φ0, X3, and 1 have structures given by
x1 0 0 1 1 1 1 0 0
1
Φ0 =  0 x2 0 , X3 =  1 1 1 , 1 =  0 1 0 . (1.2)
3
 0 0 x3   1 1 1   0 0 1 
     
The coefficients a (f = e,u,d) which are important parameters in the model play an essential
f
role in the mass ratios and mixings, while the family-number independent coefficients k and
f
ku′ do not. The values of (x1,x2,x3) with x21 +x22 +x23 = 1 are fixed by the observed charged
lepton mass values underthe given value of a . (In an earlier model [7], the charged lepton mass
e
matrix M was given by M = k′Φ Φ and M and Mˆ are given by those with the replacement
e e e e e d u
Φ0 → Φe. The structures with (1 + afX3) were suggested in a phenomenological model by
Fusaoka and one of the authors [8].)
The model has given almost successful unified description and predictions of U and
PMNS
VCKM. However, the model has failed to give the observed large mixing of θ13 in UPMNS: the
observed value is sin22θ13 ∼ 0.09 [9], while the yukawaon model predicts sin22θ13 ∼ 10−4 [1].
Since the model does not contain enough number of adjustable parameters as it is, it is hard to
improve the prediction of sin22θ13 ∼ 10−4 without the cost of other successful predictions.
An introducing of the structure X2 into the yukawaon model has been done in Ref.[2]. In
Ref.[2], thestructureX2 is introducedin Me together withasummption MD = Me, and only the
UPMNS are discussed, but the predicted value of sin22θ13 is still small: sin22θ13 ∼ 10−2. The
V was not discussed in Ref.[2]. Even in a recent revised yukawaon model [3], the predicted
CKM
2
value was, at most, sin 2θ13 ∼ 0.03.
Inthepresentmodel,itisassumedfromaphenomenological pointofviewthattheneutrino
Dirac mass matrix M is given with a somewhat different structure from the charged lepton
D
mass matrix M :
e
MD = keΦ0(1+aDX2)Φ0, (1.3)
2
where
1 1 0
1
X2 =  1 1 0 . (1.4)
2
 0 0 0 
 
Using this form we shall discuss UPMNS as well as VCKM of the model. As to the structure X2,
we will discuss in Sec.2. When once we accept the form (1.3), we predict a reasonable value of
2
sin 2θ13 ∼ 0.07 together with reasonable other parameters of UPMNS, VCKM and quark and
lepton mass ratios.
2 Model
We assume that a would-be Yukawa interaction is given as follows:
y y y y
W = eecYijℓ H + ννcYijℓ H +λ νcYijνc+ ddcYijq H + uucYijq H , (2.1)
Y Λ i e j d Λ i D j u R i R j Λ i d j d Λ i u j u
where ℓ = (ν ,e ) and q = (u ,d ) are SU(2) doublets. Under this definition of Y , the
L L L L L f
†
CKM mixing matrix and the PMNS mixing matrix are given by V = U U and U =
CKM u d PMNS
U†U , respectively, where U are defined by U†M†M U = D2 (D are diagonal). In order to
e ν f f f f f f f
distinguish each yukawaon from others, we assume that Y have different R charges from each
f
other together with R charge conservation. (Of course, the R charge conservation is broken at
the energy scale Λ′.)
We assume the following superpotential for yukawaons by introducing fields Θf, P, E, E¯,
E′, E¯′′, E′′, E¯′, φ , and φ :
e d
1 1
We = λe(cid:26)φeYeij + Λ(Φ0)iα(cid:18)(E′′)αβ +aeΛ2XαkE¯klXlTβ(cid:19)(ΦT0)βj(cid:27)Θeji, (2.2)
λ 1
WD = ΛD (cid:26)(E′)αi YDij(E′)βj +(ΦT0)αi(cid:18)Eij +aDΛ2XjTγ(E¯′′)γδXδj(cid:19)(Φ0)jβ(cid:27)ΘDβα, (2.3)
λ
W = u P YklP +YˆuE¯klYˆu Θji, (2.4)
u Λ ik u lj ik lj u
n o
λ′ 1
Wu′ = Λu (cid:26)E¯ikYˆkulE¯lj +(Φ0)iα(cid:18)(E′′)αβ +auΛ2XαkE¯klXlTβ(cid:19)(ΦT0)βj(cid:27)Θuji′, (2.5)
1 1
Wd = λd(cid:26)φdYdij + Λ(cid:20)(Φ0)iα(cid:18)(E′′)αβ +adΛ2XαkE¯klXlTβ(cid:19)(ΦT0)βj +m0d(E¯′)iα(E¯′′)αβ(E¯′)jβ(cid:21)(cid:27)Θdji,
(2.6)
3
λ
W = µ Yij + R YikYˆuE¯lj +E¯ikYˆuYlj +ξ0YikE Ylj ΘR. (2.7)
R (cid:26) R R Λ e kl kl e ν D kl D (cid:27) ji
h i
Here, we have assumed family symmetries U(3)×U(3)′. The fundamental yukawaon Φ0 is as-
signed to (3, 3) of U(3)× U(3)′, although quarks and leptons are still assigned to (3, 1) and
yukawaons Y are assigned to (6∗, 1) of U(3)× U(3)′. In order to distinguish R charges between
f
Y and Y , we have introduced U(3)×U(3)′ singlet scalar fields φ and φ . The present model is
e d e d
not anomaly freefor U(3)×U(3)′ if we consider only the fields which are given in Eqs.(2.1)-(2.7).
In order to make the model anomaly free, we need further fields. However, since roles of such
additional fields in the present model are, at present, not clear, we do not discuss such fields.
From Eqs.(2.2) and (2.3) [and also (2.5) and (2.6)], we obtain
R(E)+R(E¯) = R(E′′)+R(E¯′′). (2.8)
The VEVs of the introduced fields E, E¯, P, and P¯ are described by the following superpotential
by assuming R(EE¯) = R(PP¯)= 1:
W = λ1Tr[E¯EP¯P]+ λ2Tr[E¯E]Tr[P¯P], (2.9)
E,P
Λ Λ
which leads to
hEihE¯i ∝ 1, hPihP¯i∝ 1. (2.10)
We assume specific solutions of Eq.(2.10):
1 1
hEi = hE¯i= 1, (2.11)
v v¯
E E
1 1
hPi = hP¯i† = diag(e−iφ1,e−iφ2,1), (2.12)
v v¯∗
P P
as the explicit forms of hEi, hE¯i, and hP¯i. We assume similar superpotential forms for E′′ and
E¯′′, and for E′ and E¯′.
From SUSY vacuum conditions ∂W/∂Θ = 0, we obtain the following relations:
hYei = kehΦ0i 1+aeXXT hΦT0i, (2.13)
(cid:0) (cid:1)
hYDi = kDhΦT0i 1+aDXTX hΦ0i, (2.14)
(cid:0) (cid:1)
hPihY ihPi = k′hYˆuihYˆui, (2.15)
u u
hYˆui = kuhΦ0i 1+auXXT hΦT0i, (2.16)
(cid:0) (cid:1)
hYdi = kd hΦ0i 1+adXXT hΦT0i+m0d1 , (2.17)
(cid:2) (cid:0) (cid:1) (cid:3)
4
hY i= k hY ihYˆui+hYˆuihY i+ξ0hY ihY i , (2.18)
R R e e ν D D
(cid:16) (cid:17)
where, for convenience, we have already put hEi as 1, and so on. Here, since we have assumed
that all Θ fields take hΘi = 0, we do not need to consider vacuum conditions for other fields
∂W/∂Y = 0, because those alway contain hΘi. Thus, mass matrices are given by M = hY i,
e e e
M = hY i, M = k′Mˆ Mˆ , Mˆ = hYˆui, M = hY i, M = M M−1MT, and M = hY i.
D D u u u u u d d ν D R D R R
The most curious assumption is to assume the VEV matrix form of the scalar X as
1 1 0
1 1
hXi =  1 1 0 . (2.19)
v 2
X
 1 1 0 
 
At present, there is no idea for this form. This is only ad hoc assumption. However, as seen
2
later, we can obtain good fitting for the neutrino mixing angle sin 2θ13 due to this assumption.
The form (2.19) leads to
3 3
hXihXTi = X3, hXTihXi = X2, (2.20)
2 2
together with hXihXi = hXi, where X3 and X2 is defined by Eqs. (1.2) and (1.4), respectively,
and, for simplicity, we have put v = 1 because we are interested only in the relative ratios
X
among the family components.
3 Parameter fitting
We summarize our mass matrices in the present model as follows:
Me = keΦ0(1+aeX3)ΦT0, (3.1)
MD = kDΦT0(1+aDX2)Φ0, (3.2)
PM P = k′MˆuMˆu, (3.3)
u u
Mˆu = kuΦ0 1+aueiαuX3 ΦT0, (3.4)
(cid:0) (cid:1)
Md = kd Φ0(1+adX3)ΦT0 +m0d1 , (3.5)
(cid:2) (cid:3)
M = M M−1MT, (3.6)
ν D R D
M = k M Mˆu +MˆuM +ξ0M M , (3.7)
R R e e ν D D
(cid:16) (cid:17)
where, for convenience, we have dropped the notations “h” and “i”. Since we are interested
only in the mass ratios and mixings, hereafter, we will use dimensionless expressions Φ0 =
diag(x1,x2,x3), P = diag(eiφ1,eiφ2,1), and E = diag(1,1,1). For simplicity, we have regarded
the parameter a as real correspondingly to the parameter a . The parameters are re-refined by
d e
Eqs.(3.1)-(3.5).
5
In the present model, we have 9 adjustable parameters [ae, aD, (au,αu), ad, (φ1,φ2), m0d,
and ξ0] for the 16 observable quantities (6 mass ratios in the up-quark-, down-quark-, and
ν
neutrino-sectors, 4 CKM mixing parameters, and 4+2 PMNS mixing parameters).
Since we do not change the mass matrix structures for M , M , and M from the previous
e u d
paper [3], we use the following parameter values of a and (a ,α )
e u u
(a ,a ,α ) ∼ (8,−1.35,±8◦), (3.8)
e u u
2
which are fixed from the observed values of mc/mt, mu/mc, and sin 2θ23:
m m
ru ≡ u = 0.045+0.013, ru ≡ c = 0.06±0.005, (3.9)
12 rm −0.010 23 rm
c t
2
at µ = mZ [10], and sin 2θ23 > 0.95 [11]. (These values will be fine-tuned in whole parameter
fitting of U and V later.) Note that the neutrino sector of the model is different from
PMNS CKM
2
the previous model, however the predictrd values of sin 2θ23 are almost the same before.
atm
1
solar
10 R
0.5
t13
0
8.6 8.8 9 9.2 9.4
a
D
2 2 2
Figure1: Leptonmixingparameterssin 2θ12,sin 2θ23,sin 2θ13,andtheneutrinomasssquared
2
ratio Rν versustheparameter aD. (“solar”, “atm”, “t13”, and“10R”denotecurvesofsin 2θ12,
2 2
sin 2θ23, sin 2θ13, and Rν ×10, respectively. Other parameter values are taken as ae = 7.5,
a = −1.35, and α = 7.6◦.
u u
First, let us investigate lepton sector. In the revised model, a new parameter a is added.
D
2 2 2
We illustrate the behaviors of Lepton mixing parameters sin 2θ12, sin 2θ23, sin 2θ13, and the
neutrino mass squared ratio R versus the parameter a for a case of ξ0 = 0. As seen in Fig.1,
ν D ν
2
the parameter aD does not change the prediction sin 2θ23 ∼ 1in the previous model. Also, note
2 2
that the prediction of sin 2θ13 is insensitive to the parameter aD, i.e. sin 2θ13 ∼ 0.08. Only
6
the predictions of sin22θ12 and Rν = (m22−m21)/(m23 −m22) are sensitive to the parameter aD.
2
In order to fit the observed value [11] sin 2θ12 = 0.857±0.024, we take aD = 9.01. However, in
the model with ξ0 = 0, the value a = 9.01 cannot fit the observed value [11] of R ,
ν D ν
∆m2 (7.50±0.20)×10−5 eV2
R ≡ 21 = = (3.23+0.14)×10−2. (3.10)
ν ∆m2 (2.32+0.12)×10−3 eV2 −0.19
32 −0.08
Thenon-zeroparameterξ0 hasphenomenologically beenbroughtinordertoadjustthepredicted
ν
2 2 2
value of Rν. The predicted values of sin 2θ23, sin 2θ12, and sin 2θ13 are almost unchanged
against the parameter ξ0. In order to fit the neutrino mass ratio R , we take ξ0 = −0.78.
ν ν ν
Next, we discuss quark sector. Since we have fixed the five parameters a , a , α , a ,
e u u D
and ξ0, we have remaining four parameters for six observables (2 down-quark mass ratios and 4
ν
CKM mixing parameters). The parameters a and m0 are used to fit the observed down-quark
d d
mass ratios [10]
m m
rd ≡ s = 0.019+0.006, rd ≡ d = 0.053+0.005, (3.11)
23 m −0.006 12 m −0.003
b s
respectively. Therefore, the four CKM ming parameters are described only by two parameters
(φ1,φ2). As shown in Fig. 2, all the experimental constraints on CKM mixing matrix elements
are satisfied by fine tuning with use of only two parameters (φ1,φ2).
Finally, we do fine-tuning of whole parameter values in order to give more improved fitting
with the whole data. Our final result is as follows: under the parameter values
a = 7.5, a = 9.01, (a ,α )= (−1.35,−7.6◦), a = 25,
e D u u d
m0d = 0.0115, ξν0 = −0.78, (φ1,φ2)= (177.0◦,197.4◦), (3.12)
we obtain
ru = 0.0465, ru = 0.0614, rd = 0.0569, rd = 0.0240, (3.13)
12 23 12 23
2 2 2
sin 2θ23 = 0.969, sin 2θ12 =0.860, sin 2θ13 = 0.0711, Rν = 0.0324, (3.14)
δℓ = −131◦ (Jℓ = −2.3×10−2), (3.15)
CP
|(V ) |= 0.2271, |(V ) |= 0.0394, |(V ) |= 0.00347, |(V ) |= 0.00780,
CKM us CKM cb CKM ub CKM td
(3.16)
δq = 59.6◦ (Jq = 2.6×10−5). (3.17)
CP
Here, δℓ and δq are Dirac CP violating phases in the standard conventions of U and
CP CP PMNS
V , respectively.
CKM
2
Thepredictedvalueofsin 2θ13 = 0.0711isstillsomewhatsmallcomparedwiththeobserved
value sin22θ13 = 0.098±0.013 [11]. So far, we have assumed that the parameter ξν0 is real. If
we consider that the parameter ξν0 is complex, ξν0 → ξν0eiαν, we can adjust the value of sin22θ13
7
350
300
250 (cid:95)(cid:11)(cid:57)(cid:3)(cid:38)(cid:3)(cid:46)(cid:48)(cid:3)(cid:12)(cid:3)(cid:88)(cid:3)(cid:69)(cid:95)
(cid:12)
(cid:72)(cid:74) 200
(cid:71)
(cid:11)
(cid:557)(cid:20) 150 (cid:95)(cid:11)(cid:57)(cid:3)(cid:38)(cid:46)(cid:3)(cid:48)(cid:3)(cid:12)(cid:88)(cid:3)(cid:86)(cid:3)(cid:95)
100
(cid:95)(cid:11)(cid:57)(cid:3)(cid:38)(cid:3)(cid:46)(cid:48)(cid:3)(cid:12)(cid:70)(cid:3)(cid:69)(cid:3)(cid:95)
50 (cid:95)(cid:11)(cid:57)(cid:3)(cid:38)(cid:3)(cid:46)(cid:48)(cid:3)(cid:12)(cid:3)(cid:87)(cid:3)(cid:71)(cid:95)
0
0 50 100 150 200 250 300 350
(cid:557)(cid:11)(cid:71)(cid:72)(cid:74)(cid:12)
(cid:21)
Figure2: Contourplotsinthe(φ1,φ2)parameterplane,whichareshownbyusingexperimental
constraints on CKM mixing matrix elements: |(V ) | = 0.2252 ± 0.0009, |(V ) | =
CKM us CKM cb
0.0406±0.0013, |(V ) |= 0.00389±0.00044, and |(V ) | = 0.0084±0.0006. The CKM
CKM ub CKM td
elements depends on only the parameter set of [ae, (au, αu), ad, m0d, φ1, and φ2]. Here we
present contour plots of the CKM elements in the (φ1, φ2) parameter plane by taking the values
of other parameters as a = 7.5, a = −1.35, α = −7.6◦, a = 25, and m0 = 0.0115. We find
e u u d d
that (φ1, φ2)=(177.0◦, 197.4◦) is consistent with all the CKM constraints.
without changing other predicted values as seen in Fig.3. However, such a modification by the
parameter α is not essential in the present model, so that we will regard that the parameter ξ0
ν ν
is real.
We can also predict neutrino masses, for the parameters given (3.12) with real ξ0,
ν
mν1 ≃ 0.0048 eV, mν2 ≃ 0.0101 eV, mν3 ≃ 0.0503 eV, (3.18)
by using the input value [12] ∆m2 ≃ 0.00243 eV2. We also predict the effective Majorana
32
neutrino mass [13] hmi in the neutrinoless double beta decay as
hmi = m1(UPMNS)2e1+m2(UPMNS)2e2+m3(UPMNS)2e3 ≃ 7.3×10−4 eV. (3.19)
(cid:12) (cid:12)
(cid:12) (cid:12)
6 Concluding remarks
In conclusion, by assuming VEV matrix forms of the yukawaons Eqs.(3.1) -(3.7), we have
obtained reasonable V and U mixing parameters together with quark and neutrino
CKM PMNS
mass ratios. The major change from the previous yukawaon models is in the form of M .
D
Although we give the form by assuming the VEV matrix X which is geven by Eq.(2.19), and
by considering the mechanism XTX versus XXT, it is still phenomenological and somewhat
2
factitious. However, when once we accept the form of MD, we can obtain sin 2θ13 ∼ 0.07 whose
value is not sensitive to the other parameters.
8
atm
1
solar
10 t13
0.5
10 R
0
0 20 40
αν (cid:11)(cid:71)(cid:72)(cid:74)(cid:12)
2 2 2
Figure3: Leptonmixingparameterssin 2θ12,sin 2θ23,sin 2θ13,andtheneutrinomasssquared
ratio R versus the phaseparameter α . (“solar”, “atm”, “10 t13”, and “10 R” denote curves of
ν ν
2 2 2
sin 2θ12, sin 2θ23, sin 2θ13×10, andRν×10, respectively. Heretheαν dependenceis presented
under the parameter values given by (3.12).
Note that the present model does not have any family-dependent parameters except for
(x1,x2,x3) in hΦ0i and (φ1,φ2) in hPi. The parameter values (x1,x2,x3) have been fixed by the
observed charged lepton masses. Therefore, the model have only 9 adjustable parameters for 16
observables. The 5 parameter values of 9 parameters, (a , a , (a ,α ), and ξ0), have been fixed
e D u u ν
2 2
by the observed values mu/mc, mc/mt, sin 2θ12, sin 2θ23, and Rν. Especially, the parameter
aD has been fixed the observed value sin22θ12. The parameter ξν0 has been introduced only in
order to adjust the ratio R . (In other words, the value of ξ0 has been fixed by Robs, the value
ν ν ν
(3.10).) Logically speaking, we need four observed values in order to fix the four parameters
2 2
ae, aD, and (au,αu). However, as seen in Fig.1, the values sin 2θ23 ∼ 0.9 and sin 2θ13 ∼ 0.08
are almost determined independently of the parameter a when we fix a and (a ,α ) from the
D e u u
2 2
observed up-quark mass ratios. Therefore, sin 2θ23 ∼ 0.9 and sin 2θ13 ∼0.08 can substantially
be regarded as predictions in the present model. Of course, after we fix the 5 parameters,
2
predictions are 6 quantities: sin 2θ13, 2 neutrino mass ratios, CP-violating phase parameter,
and 2 Majorana phases.
Of the remaining 4 parameters ad, m0d, and (φ1,φ2), the parameters ad and m0d are deter-
mined by the down-quark mass ratios m /m and m /m , respectively. Therefore, the 4 CKM
s b d s
mixing parameters are predicted only by adjusting the two parameters (φ1,φ2). We can obtain
reasonable predictions of the CKM mixing parameters.
The present model is still in a level of a phenomenological model. Nevertheless, it seems
that the yukawaon model offers us a promising hint for a unified mass matrix model for quarks
9
and leptons, because quark and lepton mass matrices are given each other by their mixed terms.
Acknowledgment
One of the authors (YK) would like to thank members of the particle physics group at
NTU and NTSC, Taiwan, for the helpful discussions and their cordial hospitality. The work is
supported by JSPS (No. 21540266).
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