One-loop renormalization group equations of the neutrino mass matrix in the triplet seesaw model

0.17 MB·English

by  Wei Chao

One-loop renormalization group equations of the neutrino mass
matrix in the triplet seesaw model
Wei Chao ∗, He Zhang †
CCAST (World Laboratory), P.O. Box 8730, Beijing 100080, China
and Institute of High Energy Physics, Chinese Academy of Sciences,
P.O. Box 918, Beijing 100049, China
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Abstract
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a Withintheframeworkofthestandardmodelplusoneheavy Higgs triplet, we
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derive a full set of one-loop renormalization group equations of the neutrino

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mass matrix and Higgs couplings in both full and effective theories. The ex-
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plicit RGEs of neutrino masses, flavor mixing angles and CP-violating phases

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are also obtained, and their non-trivial running behaviors around the Higgs
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3 triplet mass threshold are numerically illustrated.
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∗E-mail: [email protected]
†E-mail: [email protected]
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I. INTRODUCTION
To understand the origin of fermion masses and flavor mixing is crucial and essential
in modern particle physics. Current neutrino experiments have provided very convincing
evidence that neutrinos are massive and lepton flavors are mixed [1–5]. However, it remains
unclear why the neutrino masses arequite suppressed compared tothe other fermionmasses.
Among various possible models, the seesaw mechanism [6] should be one of the most favorite
mechanismstoexplaintheunnaturalsmallnessoftheneutrinomassscale. IntheusualType-
I seesaw mechanism, heavy right-handed Majorana neutrinos are introduced to generate the
light neutrino masses. The lepton-flavor-violating and lepton-number-violating processes,
which are forbidden in the standard model (SM), can also take place through the exchanges
of heavy right-handed neutrinos.
Besides the Type-I seesaw scenario, the triplet seesaw mechanism [7,8], which extends
the SM with one Higgs triplet ξ = (ξ++,ξ+,ξ0)T with hypercharge 1, gives another possible
solution to the tiny light neutrino masses. Such a Higgs triplet can be introduced in many
Grand Unified Theories (GUTs) or SU(2) SU(2) theories. Note that, in general the
L × R
triplet seesaw model is more predictive than the usual Type-I seesaw model, since there is no
unknown right-handed Majorana neutrino mass matrix. The minimal version of the triplet
seesaw models contains only one triplet besides the SM particles. That is very different from
the Type-I seesaw model, in which at least two heavy right-handed Majorana neutrinos are
introduced to give rise to at least two massive left-handed neutrinos [9].
Taking into account that most of the realistic triplet seesaw models are built at some
energy scales M much higher than the typical electroweak scale M = 91.2 GeV [10], it is
NP Z
very meaningful to consider the radiative corrections to the neutrino mixing parameters. In
the energy scales below the seesaw scale, the running of the dimension-5 operator has been
considered by many authors [11] in the Type-I seesaw framework, and it has been proved
that there is no remarkable corrections in the SM or minimal supersymmetric standard
model (MSSM) with small tanβ. However, in the energy scales above the triplet seesaw
scale, the renormalization group equations (RGEs) are quite different from those in the low
energy effective theory, and the RGE analyses are still lacking in that energy scale. Since
the RGE running may bring significant corrections to the physical parameters, it is very
important for both model building and phenomenology to study the running effects from
the GUT scale (M = 1016 GeV) to the seesaw scale.
GUT
In this letter, we derive a full set of one-loop RGEs in the framework of the SM extended
with one heavy Higgs triplet. The β-functions of Yukawa couplings are calculated in detail.
The RGEs of the couplings between the triplet and doublet Higgs are also obtained. Analyt-
ical and numerical analyses based on our formulae are also given for illustration. We show
that there may be sizable radiative corrections to the mixing parameters when the triplet
Higgs is involved.
The letter is organized as follows: In section II, the basic concepts of the triplet seesaw
model is briefly discussed. In section III, we present the β-functions of Yukawa and Higgs
couplings in the model. Section IV is dedicated to the numerical analyses of the running
effects on mixing parameters. Finally, a summary is given in section V.
2
II. THE TRIPLET SEESAW MODEL
The full Lagrangian of the triplet seesaw model is given by [8]:
= + , (1)
Lfull LSM Lξ
where the first part represents the SM Lagrangian, and the second part contains the inter-
actions involving the Higgs triplet. The most general form of is
Lξ
1
= (D ξ)†(D ξ) M2(ξ†ξ) λ (ξ†ξ)2 λ (ξ†ξ)(φ†φ) λ (ξTCˆξ)†(ξTCˆξ)
Lξ µ µ − ξ − 4 ξ − φ − C
λ (ξ†t ξ)(φ†τiφ) 1 (Y ) lCfε∆lg +λ M φ˜Tε∆φ˜+h.c. , (2)
− T i 2 − 2 (cid:20) ξ fg L L H ξ (cid:21)
where φ is the SM Higgs doublet with φ˜= iτ φ∗ and ∆ is a 2 2 representation of the Higgs
2 ×
triplet field [12]:
ξ+/√2 ξ++
∆ = − . (3)
(cid:18) ξ0 ξ+/√2(cid:19)
−
In Eq. (2), f and g are generation indices, and summation over repeated indices is implied.
t are the three dimensional representations of the Pauli matrices
i
0 1 0 0 i 0 1 0 0
1 1 −
t = 1 0 1 , t = i 0 i , t = 0 0 0  , (4)
1 √2 2 √2 − 3
0 1 0 0 i 0  0 0 1
     − 
ˆ
and C is defined as [13]:
0 0 1
Cˆ =  0 1 −0  . (5)
 1 0 0 
− 
The covariant derivative D reads
µ
D ξ ∂ ξ +ig YB ξ +ig t W ξ . (6)
µ ≡ µ 1 µ 2 · µ
As already mentioned in the first section, in order to generate tiny light neutrino masses,
the mass scale of the Higgs triplet M should be much higher than the typical electroweak
ξ
scale M . In this letter, we take M = 1010 GeV. In the energy scale µ M , the full
Z ξ ≫ ξ
Lagrangian is taken into account. In the low energy limit µ M , we should use the
≪ ξ
effective theory by integrating out the heavy triplet field. The effective Lagrangian can be
defined by [14] :
3
exp i d4x (x) ξ ξ† exp i d4x (x)
(cid:26) Z Leff (cid:27) ≡ Z D iD i (cid:26) Z Lfull (cid:27)
Yi
3
= exp i d4x (x) ξ ξ† exp i d4x (x) , (7)
(cid:26) Z LSM (cid:27)Z D iD i (cid:26) Z Lξ (cid:27)
Yi
3
where ξ stands for functional integration over ξ . Starting from Eq. (7), we can calculate
D i i
the one-loop level effective Lagrangian by using the steepest-descent method to integrate
out the heavy scalar. Keeping only terms of order (1/M2) through the whole calculation
O ξ
and neglecting all the operators with higher inverse powers of M , we may get the effective
ξ
operators:
1
= λ† λ (φ†φ)2 ;
L4−Higgs −4 H H
= 1λH(Yξ)fg(lCfεφ)(lgεφ)+h.c. ;
Lν−mass −4 M L L
ξ
= 1(Yξ†)mn(Yξ)fg (lm lg)(lCf lCn)+(lCf lmT)(lgT lCn) , (8)
L4−Fermi −8 M2 (cid:20) L · L L · L L · L L · L (cid:21)
ξ
where m,n,f and g run over 1,2,3. has the same form as the Higgs self-coupling in
LHiggs
the SM. The four fermion interaction may contribute to the lepton flavor violating
L4−Fermi
processes. However, such processes should be strongly suppressed due to the heavy mass of
the Higgs triplet. Thus we will neglect this four fermion coupling in the following calcula-
tions. is in proportion to the Majorana neutrino mass matrix. In analogy to the
Lν−mass
Type-I seesaw model, we can also define the effective dimension-5 operator
= 1κ (lCfεφ)(lgεφ)+h.c. , (9)
Lν−mass −4 fg L L
where κ = λ Y /M . After spontaneous symmetry breaking, neutrinos acquire masses
H ξ ξ
and the neutrino mass matrix is given by M = Y ∆0 with ∆0 = λ v2/M . Here
ν ξh i h i H ξ
v 174 GeV denotes the Higgs vacuum expectation value. Since M v, the mass scale
≃ ξ ≫
of neutrinos is then suppressed, and this is the so-called triplet seesaw mechanism.
III. CALCULATIONS OF THE BETA-FUNCTIONS
In our calculations, we always use the dimensional regularization. For the one-loop
wavefunction renormalization constants Z above the seesaw scale, we find that
1 3 1 3 1
δZ = Y†Y +Y†Y + g2ξ + g2ξ ,
lL −16π2 (cid:18)2 ξ ξ e e 2 1 B 2 2 W(cid:19) ε
1 1 3 1
δZ = 2Tr Y†Y +3Y†Y +3Y†Y + g2(ξ 3)+ g2(ξ 3) ,
φ −16π2 (cid:20) (cid:16) e e d d u u(cid:17) 2 1 B − 2 2 W − (cid:21) ε
1 1
δZ = Tr(Y†Y )+2g2(ξ 3)+4g2(ξ 3) . (10)
ξ −16π2 h ξ ξ 1 B − 2 W − i ε
For the vertex renormalization constants and the Higgs masses, we obtain
1 1 1 1
δZ = g2(3 3ξ )+ g2(3 7ξ ) ,
Yξ 16π2 (cid:20)2 1 − B 2 2 − W (cid:21) ε
1 3 7 1
δλ = g2ξ + g2ξ ,
H −16π2 (cid:20)2 1 B 2 2 W(cid:21) ε
1 1 1
δm2 = 3λm2 +6λ m2 +3λ† λ m2 (g2ξ +3g2ξ )m2 ,
φ 16π2 (cid:20) φ φ ξ H H ξ − 2 1 B 2 W φ(cid:21) ε
1 1
δm2 = 4λ m2 +2λ m2 +4λ m2 +λ† λ m2 2(g2ξ +2g2ξ )m2 . (11)
ξ 16π2 h ξ ξ C ξ φ φ H H ξ − 1 B 2 W ξi ε
4
The counterterms of the Higgs couplings λ , λ , λ and λ have also been calculated,
ξ C φ T
1
δλ = 6λ2 +2λ2 +8λ2 +4λ2 +4λ λ 4g2λ 8g2λ +24g4+72g4
ξ 16π2 n ξ T φ C ξ C − 1 ξ − 2 ξ 1 2
1
+48g2g2+Tr[(Y†Y )2] ,
1 2 ξ ξ o ε
1 1
δλ = 3λ2 +6λ λ 2λ2 4g2λ 8g2λ 48g2g2 2Tr[(Y†Y )2] ,
C 16π2 n C ξ C − T − 1 C − 2 C − 1 2 − ξ ξ o ε
1 5 11 1
δλ = 4λ2 +2λ2 +4λ λ +2λ λ +3λ λ g2λ g2λ +3g4 +9g4 ,
φ 16π2 (cid:18) φ T ξ φ C φ φ − 2 1 φ − 2 2 φ 1 2(cid:19) ε
1 5 11 1
δλ = 8λ λ +λ λ 2λ λ +λλ g2λ g2λ +12g2g2 . (12)
T 16π2 (cid:18) φ T ξ T − C T T − 2 1 T − 2 2 T 1 2(cid:19) ε
By using the counterterms calculated above and the technique described in [11], we
obtain the β-functions (β = µ d X) of Yukawa couplings and λ :
X dµ H
3 1 3 1
16π2β = Y (Y†Y ) + (Y†Y ) + (Y†Y )T + (Y†Y )T Y
Yξ ξ(cid:20)4 ξ ξ 2 e e (cid:21) (cid:20)4 ξ ξ 2 e e (cid:21) ξ
1
+ Tr Y†Y 3g2+9g2 Y ,
2 h (cid:16) ξ ξ(cid:17)−(cid:16) 1 2(cid:17)i ξ
3 3 15 9
16π2β = Y Y†Y + Y†Y +Tr(Y†Y +3Y†Y +3Y†Y ) g2 g2 ,
Ye e(cid:20)4 ξ ξ 2 e e e e u u d d − 4 1 − 4 2(cid:21)
1 1
16π2β = λ Tr Y†Y +2Y†Y +6Y†Y +6Y†Y + λ ( 9g2 21g2) , (13)
λH H (cid:18)2 ξ ξ e e u u d d(cid:19) 2 H − 1 − 2
and the anomalous dimensions (γ = 1µdm) of the Higgs masses:
m −m dµ
3 3 9 1 m2
16π2γ = λ+Tr Y†Y +3Y†Y +3Y†Y + g2 + g2 3(λ + λ† λ ) ξ ,
mφ −(cid:20)2 (cid:16) e e u u d d(cid:17)(cid:21) 4 1 4 2 − φ 2 H H m2
φ
1 1 m2
16π2γ = 2λ +λ + λ† λ + Tr Y†Y 3g2 6g2 2λ φ . (14)
mξ −(cid:20) ξ C 2 H H 2 (cid:16) ξ ξ(cid:17)− 1 − 2(cid:21)− φm2
ξ
It should be noticed that in the triplet seesaw model the mass of the Higgs doublet suffers
fromtheso-calledhierarchy problem, which canbeprevented insomeothersupersymmetric
models. We also calculate the β-functions of Higgs couplings:
3 2
16π2β = 6λ2 +12λ2 +2λ2 3λ g2 +3g2 +3g4 + g2 +g2
λ φ T − (cid:16) 1 2(cid:17) 2 2 (cid:16) 1 2(cid:17)
+4λTr Y†Y +3Y†Y +3Y†Y
e e u u d d
(cid:16) (cid:17)
8Tr (Y†Y )2 +3(Y†Y )2 +3(Y†Y )2
− h e e u u d d i
16π2β = 2λ2 +8λ2 +4λ λ +4λ2 +6λ2 12g2λ 24g2λ +24g4
λξ T φ ξ C C ξ − 1 ξ − 2 ξ 1
+72g4+48g2g2+4Tr (Y†Y )2 +2λ Tr Y†Y
2 1 2 ξ ξ ξ ξ ξ
h i (cid:16) (cid:17)
16π2β = 3λ2 +6λ λ 2λ2 12g2λ 24g2λ 48g2g2
λC C ξ C − T − 1 C − 2 C − 1 2
+2λ Tr Y†Y 2Tr (Y†Y )2
C (cid:16) ξ ξ(cid:17)− h ξ ξ i
5
15 33
16π2β = 4λ2 +2λ2 +4λ λ +2λ λ +3λλ g2λ g2λ +9g4
λφ φ T ξ φ C φ φ − 2 1 φ − 2 2 φ 2
1
+3g4+2λ Tr Y†Y +Y†Y +3Y†Y +3Y†Y
1 φ (cid:18)2 ξ ξ e e d d u u(cid:19)
15 33
16π2β = 8λ λ +λ λ 2λ λ +λλ g2λ g2λ +12g2g2
λT φ T ξ T − C T T − 2 1 T − 2 2 T 1 2
1
2λ Tr Y†Y +Y†Y +3Y†Y +3Y†Y . (15)
− T (cid:18)2 ξ ξ e e d d u u(cid:19)
The RGEs for the gauge couplings are changed in this model, and we list the results below:
47
16π2β = g3 ,
g1 6 1
5
16π2β = g3 ,
g2 −2 2
16π2β = 7g3 . (16)
g3 − 3
Since the Higgs triplet field does not couple with quarks, the RGEs of Yukawa couplings for
quarks are the same as those in the SM, and the corresponding results can be found in the
literature [15].
By calculating the relevant one-loop diagrams, we obtain the β-function of the effective
operator which describes the neutrino masses and mixing at the energy scales below the
seesaw scale:
3 3
16π2β = κ(Y†Y )T (Y†Y )κ+(λ+λ† λ )κ 3g2κ
κ −2 e e − 2 e e H H − 2
+2Tr Y†Y +3Y†Y +3Y†Y κ . (17)
e e u u d d
(cid:16) (cid:17)
It should be mentioned that Eq. (17) has the same form as that in the Type-I seesaw model,
only up to a replacement λ λ+λ† λ .
→ H H
IV. APPLICATIONS
To see the running behaviors of neutrino mixing parameters in the triplet seesaw model,
we carry out some numerical analyses by using the β-functions derived above. The lepton
flavor mixing matrix, which comes from the mismatch between the diagonalizations of the
neutrino mass matrix and the charged lepton mass matrix, is given by V = V†V , where
e ξ
V† (Y†Y ) V = Diag y2, y2, y2 ,
e · e e · e (cid:16) e µ τ(cid:17)
VT Y V = Diag(y , y , y ) , (18)
ξ · ξ · ξ 1 2 3
with (y , y , y ) and (y , y , y ) being the eigenvalues of Y and Y respectively. In this
e µ τ 1 2 3 e ξ
letter, we adopt the following parameterization [16]:
c c s c s eiρ 0 0
12 13 12 13 13
V =  c s s s c e−iδ s s s +c c e−iδ s c  0 eiσ 0 , (19)
− 12 23 13 − 12 23 − 12 23 13 12 23 23 13
 c c s +s s e−iδ s c s c s e−iδ c c  0 0 1
− 12 23 13 12 23 − 12 23 13 − 12 23 23 13 
6
where c cosθ and s sinθ (for ij = 12,23 and 13).
ij ≡ ij ij ≡ ij
For the typical choice ∆0 (0.1 eV), one can estimate that Y (1), which
h i ∼ O ξ ∼ O
means that the relation Y Y holds, and the tiny Yukawa coupling Y in the RGEs of Y
ξ ≫ e e ξ
and Y can be safely neglected in the hierarchical mass spectrum case 1. Then we get the
e
approximate equations of Yukawa couplings:
dY 1 3
µ ξ Y Y†Y +α ,
dµ ≃ 16π2 ξ(cid:18)2 ξ ξ ξ(cid:19)
dY 1 3
µ e Y Y†Y +α , (20)
dµ ≃ 16π2 e(cid:18)4 ξ ξ e(cid:19)
where
1
α = Tr Y†Y 3g2+9g2 ,
ξ 2 h (cid:16) ξ ξ(cid:17)−(cid:16) 1 2(cid:17)i
15 9
α = Tr Y†Y +3Y†Y +3Y†Y g2 g2 . (21)
e (cid:16) e e u u d d(cid:17)− 4 1 − 4 2
Using the results above and taking into account the fact that m > m m , we neglect
τ µ ≫ e
the tiny terms in proportion to y2 and arrive at the approximate analytical results of three
e
mixing angles:
dθ 1 3
µ 12 s c ∆ , (22)
dµ ≃ −16π2 · 4 12 12 21
dθ 1 3
µ 23 2cosδs c c2 s s c (s2 c2 s2 ) ∆ s c c2 ∆ , (23)
dµ ≃ 16π2 · 4 nh 12 12 23 13 − 23 23 12 − 12 13 i 21 − 23 23 13 32o
dθ 1 3
µ 13 s c c2 ∆ +∆ , (24)
dµ ≃ −16π2 · 4 13 13(cid:16) 12 21 32(cid:17)
where ∆ = y2 y2 with (i,j = 1,2,3). Note that, in deriving Eqs. (22)-(24), we have
ij i − j
adopted the the parametrization in Eq. (19). Such instructive expressions allow us to do
useful analyses of the running behaviors of mixing angles. Due to the hierarchical charged
leptonmasses, thereisingeneralnoenhancedfactorcomparedwiththeType-Iseesaw model
[11]. However, nontrivial running effects may also be acquired from the sizable Yukawa
coupling Y . From Eqs. (22)-(24), one can immediately conclude that the corrections to θ
ξ 12
and θ should be milder than that to θ since the right-hand sides of Eqs. (22) and (24)
13 23
are in proportion to either ∆ or θ . In the limit θ 0 and ∆ 0, we can see from
˙ 21 13 13 → 21 →
Eq. (23) that θ ∆ . Thus θ will get negative correction in the normal hierarchy
23 ∝ − 32 23
case. For illustration, we only show the evolution of θ with different λ (M ) in Fig. 1.
23 H ξ
We can see that its running is quite sensitive to λ and a decrease of several degrees may
H
be acquired from the RGE evolution.
1When the neutrino mass spectrum is nearly degenerate m m m , such an approximation
1 2 3
≃ ≃
may not bereasonable. However, in our numerical calculations, we use the exact RGEs and do not
make any approximations.
7
Considering the smallness of θ , the evolution of the Dirac CP-violating phase δ is the
13
same as those of two Majorana phases (ρ, σ) at the leading order of s−1,
13
dδ dρ dσ 1 3y2(y2 y2) sinδs c s c
µ µ µ e τ − µ 12 12 23 23∆ + (θ ) . (25)
dµ ≃ dµ ≃ dµ ≃ 16π2 · 2(y2 y2)(y2 y2) s 21 O 13
τ − e µ − e 13
This is an interesting feature: once three CP-violating phases are same at certain energy
scale, they will keep this equality against the RGE running. We can also see that the small
sinθ in the denominator of Eq. (25) dominates the running of CP phases. That means a
13
fixed point [17] should exist for extremely tiny θ . As an example, we plot the evolution of
13
δ with different θ in Fig. 2. Similar results can be obtained for two Majorana phases ρ
13
and σ.
By using Eq. (20), we obtain the RGEs of the eigenvalues of Y
ξ
dy 1 3
µ i y3 +α y , (26)
dµ ≃ 16π2 (cid:18)2 i ξ i(cid:19)
with i = 1,2,3. Note that, for different signs of α , the corrections to y may be either
ξ i
positive or negative. However, in order to investigate the running of light neutrino masses,
oneshouldconsider theRGEsofm andλ simultaneously. InFig. 3, wepresent thetypical
ξ H
evolution of three light neutrino masses with λ (M ) = 5 10−5. We can see that their
H ξ ×
running effects are appreciable and should not be neglected. A detailed numerical analysis
of the triplet seesaw model is worthwhile and the corresponding work will be elaborated
elsewhere.
V. SUMMARY
Working in the framework of the SM extended with one heavy Higgs triplet, we have
derived a full set of one-loop RGEs for lepton Yukawa and Higgs couplings. Since the triplet
seesaw model involves more couplings than the usual Type-I seesaw models, the results
are also quite different. Analytical and numerical analyses have been given based on the
RGEs we obtained. We find that nontrivial corrections to the mixing parameters can be
acquired and they should not be neglected in general. It provides us a possible way to
connect the experimental values of lepton flavor mixing parameters with some high energy
GUT theories. In conclusion, our formulae are very important for both model building and
phenomenological analyses of the triplet seesaw models.
ACKNOWLEDGMENTS
The authors are indebted to Professor Zhi-zhong Xing for reading the manuscript with
great care and patience, and for his valuable comments and numerous corrections. They
are also grateful to S. Zhou for useful discussions. This work was supported in part by the
National Nature Science Foundation of China.
8
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9
FIGURES
θ /◦
23
45.5
45
44.5
44
43.5
λ (M )=1×10−5
43 H ξ
λ (M )=2×10−5
H ξ
42.5
λ (M )=5×10−5
H ξ
42
0 5 10 15 20
10 10 10 10 10
µ (GeV)
FIG. 1. The evolution of θ . We take δ = ρ= σ = 90◦ and θ = 0.01◦ at the scale µ = M .
23 13 Z
δ/◦
93
θ (M )=0.1°
92 13 Z
θ (M )=0.01°
13 Z
91 θ (M )=0.001°
13 Z
90
89
88
87
86
0 5 10 15 20
10 10 10 10 10
µ (GeV)
FIG. 2. Examples of the evolution of the Dirac CP-violating phase δ. We take δ = 90◦ and
m =0.01 eV at the M scale. We also take λ (M )= 5 10−5. Similar results can be obtained
1 Z H Z ×
for two Majorana phases ρ and σ.
10