Fermion Mass Hierarchy in Lifshitz Type Gauge Theory

0.17 MB·English

by  Kunio Kaneta

Fermion Mass Hierarchy
in Lifshitz Type Gauge Theory
0
1
0
2
Kaneta Kawamura
n Kunio ∗and Yoshiharu †
a Department of Physics, Shinshu University,
J
Matsumoto 390-8621, Japan
8
2

January 27, 2010
]
h
p
-
p
e
h Abstract
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We study the origin of fermion mass hierarchy and flavor mix-
2
v ing in a Lifshitz type extension of the standard model including an
0 extra scalar field. We show that the hierarchical structure can orig-
2
inate from renormalizable interactions. In contrast to the ordinary
9
2 Froggatt-Nielsen mechanism, the higher the dimension of associated
.
9 operators,theheavierthefermionmasses. Tinymassesforleft-handed
0 neutrinos are obtained without introducing right-handed neutrinos.
9
0
: The origin of fermion mass hierarchy and flavor mixing is one of the
v
biggest problem in particle physics. In the standard model (SM), the hier-
i
X
archical structure originates from the texture of Yukawa couplings. Because
r
a the Yukawa couplings are free parameters, their values should be determined
by a theory beyond the SM. Hence the structure of Yukawa couplings can
give us valuable clues for exploring an underlying theory.
Recently, an exotic theory beyond the SM and/or the minimal supersym-
metric SM (MSSM) has been proposed.[1] The candidate theory is a Lifshitz
type extension of the SM and/or the MSSM.1 This type of theory is assumed
∗E-mail: [email protected]
†E-mail: [email protected]
1ALifshitztypeextensionofgravitytheorywasproposedbyHoˇrava.[2,3,4]Properties
of Lifshitz type field theory have been investigated in Ref. [5, 6, 7, 8, 9, 10].
1
to have a fixed point with anisotropic scaling characterized by a dynamical
critical exponent z(> 1) above a high-energy scale M . The system does
ℓ
not possess the relativistic invariance for z = 1. The Lorentz invariance is
6
expected to emerge after the transition from z = 1 to z = 1 around M .2
ℓ
6
In this letter, we study the origin of fermion mass hierarchy and flavor
mixing in a Lifshitz type extension of the SM including an extra scalar field.
We show that the hierarchical structure can originate from renormalizable
interactions.
The basic idea is as follows. The Lifshitz type theory can be renormaliz-
ablebypowercounting, eventhoughitcontainshigher-dimensional operators
which make the theory with z = 1 non-renormalizable. Operators of dimen-
sionality 4 + r (r > 0), O(4+r), become irrelevant ones O(4+r)/Mr after the
ℓ
transition from z = 1 to z = 1 (and the dimensional reduction if extra di-
6
mensions exist) around M . The contributions from these operators are, in
ℓ
general, negligibly small if M is sufficiently large. For example, M should
ℓ ℓ
be larger than O(1015∼16) GeV in order to suppress proton decay processes.
Suppose that a symmetry is broken down spontaneously at a high-energy
scale M larger than M and O(4+r) change into M4+r−qO(q). Then these
SB ℓ SB
operators can be relevant ones such as (M /M )4+r−qM4−qO(q) for q 4
SB ℓ ℓ ≤
below M . Here, we assume that renormalizable terms including parameters
ℓ
with positive mass dimensions originate from a specific dynamics character-
ized by a scale M and every parameter with mass dimension is given by a
ℓ
power of M . In this case, the hierarchy among couplings related to O(q) can
ℓ
originate from the difference of exponents in (M /M )4+r−q. If we apply it
SB ℓ
to the Yukawa couplings, we find an interesting feature that the higher the
dimension of associated original operators, the heavier the fermion masses.3
For the sake of completeness, we explain the ordinary Froggatt-Nielsen
mechanism[16] for our framework. When we suppose that the Lifshitz type
theoryisaneffectiveonederivedfromanunderlyingtheory,non-renormalizable
terms can appear after integrating out superheavy fields. That is, operators
ofdimensionalityD+z+p, O(D+z+p), canbederived withthesuppression fac-
tor Λp in the Lifshitz type theory on D+1-dimensional space-time, where Λ
is a cut-off scale or a mass scale related to superheavy fields. After a symme-
2 In Ref. [11, 12, 13], properties and renormalizability for quantum field theories
with Lorentz symmetry breaking terms have been studied intensively on the basis of
“weighted power counting”. Furthermore, extensions of the SM have been proposed for
this framework.[14, 15]
3 This feature changes into an opposite one if M is smaller than M .
SB ℓ
2
try breaking at M smaller than Λ, O(D+z+p) change into (M /Λ)pO(D+z)
SB SB
and the hierarchy among couplings related to O(D+z) can originate from the
difference of exponents in (M /Λ)p.4 In this case, it is known that the
SB
higher the dimension of associated original operators, the lighter the fermion
masses.
First, let us explain the fermion mass hierarchy and the flavor mixing. In
the SM, the Yukawa interactions for quarks and charged leptons are given
by
(u) (d) (e)¯
= f q¯ h u +f q¯ hd +f l he +h.c. , (1)
LY ij Li u Rj ij Li Rj ij Li Rj
(X)
where f (X = u,d,e) are the Yukawa couplings, i,j are family indices
ij
(i,j = 1,2,3), q¯ are Hermitian conjugates of left-handed quark doublets,
Li
u are right-handed up type quark singlets, d are right-handed down type
Ri Ri
¯
quark singlets, l are Hermitian conjugates of left-handed lepton doublets,
Li
e are right-handed electron type lepton singlets, h (or h iτ h∗) is a weak
Ri u 2
≡
Higgs doublet and h.c. represents Hermitian conjugates of former terms.
Quark masses and charged lepton masses are the eigenvalues of fermion mass
matrices M given by
X
v v v
(u) (d) (e)
(M ) = f , (M ) = f , (M ) = f , (2)
u ij ij √2 d ij ij √2 e ij ij √2
where v(= 246GeV) is the vacuum expectation value (VEV) of neutral com-
ponent (h0) of h.5 Using unitary matrices S and T , M are diagonalized
X X X
as
S†M T = diag(m ,m ,m ) , S†M T = diag(m ,m ,m ) ,
u u u u c t d d d d s b
S†M T = diag(m ,m ,m ) . (3)
e e e e µ τ
The quark flavor mixing is given by the Kobayashi-Maskawa matrix V =
KM
S†S .[17]Experimental data[18]showtheexistence offermionmasshierarchy
u d
and flavor mixing. Because the flavor structure originates from the texture
of Yukawa couplings and the Yukawa couplings are free parameters in the
4 Strictly speaking, an extra factor such as (M /Λ)γp appears after the dimensional
ℓ
reduction and the redefinition of fields where γ is a constant related to z and the dimen-
sionality of extra space. See (19).
5 IntheMSSM,v/√2isreplacedbythecorrespondingone,i.e.,eitherVEVforneutral
components (h0,h0) of two Higgs doublets.
u d
3
SM, we need a theory beyond the SM to strip the structure of its aura of
mystery. Suppose that an underlying theory holds above O(1015∼16) GeV.
Considering renormalization effects in the SM or the MSSM, the magnitude
of each fermion mass and each entry in V at O(1016) GeV can be roughly
KM
represented as
v v
(m ,m ,m ) (λ7,λ4,1) , (m ,m ,m ) (λ6,λ4,λ2) ,
u c t d s b
∼ √2 ∼ √2
v
(m ,m ,m ) (λ7,λ4,λ2) (4)
e µ τ
∼ √2
and
1 λ λ3
(V ) λ 1 λ2 , (5)
KM ij  
∼
λ3 λ2 1
 
 
where we use the Cabibbo angle λ sinθ 0.23.[19] For V , recall the
C KM
≡ ∼
Wolfenstein parameterization.[20] Our present goal is to derive the structure
(4) and (5) using a specific theory beyond the SM. An exotic candidate
beyond the SM is a Lifshitz type extension of the SM or the MSSM.
Next let us explain a framework of Lifshitz type field theory [1, 2, 3,
4, 5, 6, 7, 8, 9, 10] briefly. Space-time is assumed to be factorized into a
product of 3-dimensional Euclidean space R3, extra n-dimensional compact
space and time R, whose coordinates are denoted by xi (i = 1,2,3), yk
(k = 1, ,n) and t. The notation xI (I = 1, ,n+3) is also used for the
··· ···
(n + 3)-dimensional space coordinates. The dimensions of xi, yk and t are
defined as
[xi] = [yk] = 1 , [t] = z , (6)
− −
where z is the dynamical critical exponent, which characterizes anisotropic
scaling xi bxi, yk byk and t bzt at the fixed point. The system
→ → →
does not possess the relativistic invariance for z = 1 but it possesses spatial
6
rotational invariance and translational invariance. The kinetic terms for a
complex scalar field Φ and a spinor field Ψ are given by
2
∂Φ ∂
dtd3xdny +Ψ¯iΓ0 Ψ+ , (7)
(cid:12) ∂t (cid:12) ∂t ···
Z (cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12) 
(cid:12) (cid:12)
4
where Γ0 corresponds to the time component of the gamma matrices and the
ellipsis stands for terms including spatial derivatives. Ψ is a spinor defined
on Rn+3 and it transforms as
Ψ(x) Ψ′ Ψ′(x′) = S(O)Ψ(x) , (8)
→ →
i
S(O) e−4iωIJΣIJ , ΣIJ ΓIΓJ ΓJΓI , (9)
≡ ≡ 2 −
(cid:16) (cid:17)
under the spatial rotation xI x′I = OIxJ. Here, the ΓI are gamma
→ J
matrices, the ω are parameters related to the rotation angles θI with
IJ
ω = ε θK and OI is the orthogonal matrix given by OI = (eω)I. S(O)
IJ − IJK J J J
satisfies the following relations:
S†(O)ΓIS(O) = OIΓJ , S†(O)S(O) = , (10)
J I
where is the unit matrix. Chiral fermions on R3 are assumed to appear af-
I
terthedimensional reduction, e.g., throughtheorbifoldbreakingmechanism.
The engineering dimensions of Φ and Ψ are given by
3+n z 3+n
[Φ] = − , [Ψ] = , (11)
2 2
respectively. Then the dimension of operator Ψ†ΦNΨ is given by
3+n z
[Ψ¯ΦNΨ] = − N +3+n . (12)
2
The operator Ψ¯ΦNΨ becomes relevant or a renormalizable term if its di-
mension is less than or equals to 3 + n + z. The operator including higher
spatial derivatives such as Φ† 2zΦ can also become relevant. The theory
∇
can be renormalizable by power counting, even though it contains higher-
dimensional operatorswhich make thetheorywithz = 1 non-renormalizable.
The Lorentz invariance is expected to emerge as an accidental symmetry af-
ter the transition from a high-energy theory with z = 1 to that with z = 1
6
around M .6 The magnitude of Lorentz symmetry breaking terms is esti-
ℓ
mated and it gives constraints on parameters.[23, 24, 25, 26],[1]
Now let us explore the origin of texture of Yukawa couplings in a Lifshitz
type extension of the SM including an extra scalar field Φ.7 We introduce an
6 TherehasbeenaproposalthattheLorentzinvarianceappearsatanattractiveinfra-
red fixed point.[21, 22]
7 The study based on the Lifshitz type extension of the MSSM can be carried out by
the introduction of two Higgs doublets and similar results can be obtained.
5
extra U(1) symmetry denoted by U(1) and assume that Φ is a singlet under
A
the SM gauge group SU(3) SU(2) U(1) but has a non-zero U(1)
C L Y A
× ×
charge and U(1) is spontaneously broken down by the non-vanishing VEV
A
of Φ above Mℓ. The interactions such as Φm(ijX)Ψ¯iH(u)Ψj are determined by
(X)
U(1) . Here, m are zero or positive integers, and Ψ and H are fermions
A ij i (u)
and a boson which contain SM fermions ψ and a weak Higgs boson h as
i (u)
zero modes, respectively. The origin of extra U(1) symmetry is not specified
in our analysis for simplicity.8 The action is given by
1
dtd3xdny D Φ 2 Φ†(D†D )zΦ C D Φ 2
| t | − κ2 I I − Φ| I |
Z (cid:20)
1
+ D H 2 H†(D†D )zH C D H 2 +Ψ¯ iΓ0D Ψ
| t | − κ2 I I − H | I | i t i
h
−ξ1i2Ψ¯i(iΓIDI)zΨi −CΨiΨ¯iiΓIDIΨi +γi(jX)Φm(ijX)Ψ¯iH(u)Ψj +···# , (13)
where κ2, κ2 and ξ2 are dimensionless parameters concerning Lorentz sym-
h i
metry violating terms, D and D are covariant derivatives and the ellipsis
t I
stands for other terms. The engineering dimensions of fields (Φ,H ,Ψ ) and
(u) i
(X)
parameters (C ,C ,C ,γ ) are given by
Φ H Ψi ij
3+n z 3+n
[Φ] = [H ] = − , [Ψ ] = (14)
(u) i
2 2
and
[C ] = [C ] = 2(z 1) , [C ] = z 1 ,
Φ H − Ψi −
(X)
(3+n z)(m +1)
[γ(X)] = z − ij , (15)
ij − 2
respectively. We assume that renormalizable terms including parameters
with positive mass dimensions originate from a specific dynamics character-
ized by a scale M and the parameters are given by a power of M , which
ℓ ℓ
is similar to the soft supersymmetry (SUSY) breaking parameters in SUSY
models.9 The magnitude of parameters is not necessarily O(1) in the unit
8 If U(1) is an anomalousgaugesymmetry, we need to introduce extra fields in order
A
for the theory to be harmless.
9 For example, the soft supersymmetry breaking terms are given by a power of the
gravitino mass m3/2 in the gravity mediation.
6
of M but can be much smaller like most A terms in SUSY models. On the
ℓ
other hand, we assume that parameters in non-renormalizable terms are sup-
pressed by a power of cutoff scale Λ as usual. To become relativistic below
M , finetuning among parameters is required such as C = C = C2 for all
ℓ Φ H Ψi
species. In this setup, parameters are expressed as
C = C = M2(z−1) , C = Mz−1 ,
Φ H ℓ Ψi ℓ
γ0(X)Mz−(3+n−z)(2m(ijX)+1) z (3+n−z)(m(ijX)+1) ,
γi(jX) =  γi0j(X)Λzℓ−(3+n−z)(2m(ijX)+1) z ≥< (3+n−z)(2m(ijX)+1)! , (16)
ij 2
!
after a suitable rescaling of fields. Here, γ0(X) is a dimensionless parameter.
ij
We assume that the volume of extra n-dimensional space is 1/Mn. After the
ℓ
redefinition of time variable and fields as
z−n−1
x Mz−1t , Φ˜ M 2 Φ = φ+ ,
0 ≡ ℓ ≡ ℓ ···
˜ z−n−1 ˜ −n
H M 2 H = h + , Ψ M 2Ψ = ψ + (17)
(u) ≡ ℓ (u) (u) ··· i ≡ ℓ i i ···
and the dimensional reduction of extra dimensions, the following action for
zero modes is derived from (13),
m(X)
φ ij
d4x D h 2 +ψ¯iγµD ψ +γ0(X) h i ψ¯h ψ + , (18)
Z | µ | i µ i ij Mℓ! i (u) j ···
 
 
wheretheellipsis standsfortheYukawa interactionsfromnon-renormalizable
terms and so on. The dimensions of φ, h and ψ are [φ] = [h ] = 1 and
(u) i (u)
[ψ ] = 3/2. The Yukawa couplings are given by10
i
m(X)
γ0(X) hφi ij z (3+n−z)(m(ijX)+1) ,
fi(jX) =  γi0j(X) MMℓℓ!3+n2−3z Mℓ 1+n2−z hφi m(ijX) z ≥< (3+n−z)(2m(ijX)+1)! . (19)
10 Thecontijribut(cid:18)ionΛs f(cid:19)rom volum(cid:18)e sΛup(cid:19)pression fΛactor can al so appear after2the dim!en-
sional reduction.[27] In this case, the difference of field configurations related to interac-
tions can be important to study the origin of mass hierarchy. Here, we do not consider
them for simplicity.
7
(X)
The exponents m are determined from the U(1) charge conservation:
ij A
(u)
m Q (φ)+Q (q¯ )+Q (u )+Q (h ) = 0 , (20)
ij A A Li A Rj A u
(d)
m Q (φ)+Q (q¯ )+Q (d )+Q (h) = 0 , (21)
ij A A Li A Rj A
(e) ¯
m Q (φ)+Q (l )+Q (e )+Q (h) = 0 , (22)
ij A A Li A Rj A
where Q represents the charge of U(1) . The first one in (19) comes from
A A
renormalizable terms and the ratio M / φ can play a role of λ, in the case
ℓ
h i
0(X)
that there is no hierarchy among each entry in γ and φ is larger than
ij h i
M . In other words, there is a build-in mechanism to generate the hierarchy
ℓ
using the ratio M / φ on the basis of relevant operators. On the other
ℓ
h i
hand, the second one in (19) comes from non-renormalizable terms which
might originate from some renormalizable interactions after integrating out
superheavy fields. The mechanism to generate the hierarchy using the ratio
1+n−z
(Mℓ/Λ) 2 φ /Λ is regarded as the Lifshitz type extended version of the
h i
Froggatt-Nielsen mechanism.11 There is a possibility that the hierarchy of
Yukawa couplings stems from the mixture of first and second ones.
Next we consider the case with z = 4 and n = 2 as an example. The
Yukawa couplings are given by
m(X)
φ ij
0(X) (X)
γ h i (0 m 7) ,
fi(jX) =  i0j(X) MΛℓ!72 φ m(ijX) ≤(X) ij ≤ (23)
γ h i (m > 7) .
If M / φ λ,theifjerm(cid:18)ioMnℓm(cid:19)as s √hieMraℓΛrc!hy (4) canibje derived from the U(1)
ℓ A
h i ∼
charge assignment:12
Q (q¯ ) = (0,1,3) , Q (u ) = (0,2,4) , Q (d ) = (1,2,2) ,
A Li A Ri A Ri
¯
Q (l ) = (0,1,1) , Q (e ) = (0,2,4) , Q (h ) = 0 ,
A Li A Ri A (u)
Q (φ) = 1 , (24)
A
−
11 We need some selection rule related to interactions in order for the Froggatt-Nielsen
mechanism to work. In most cases, one uses an anomalous U(1) gauge symmetry,[28,
29] whose anomalies are canceled via the Green-Schwarz mechanism,[30] motivated by
superstring theories. A discrete horizontal symmetry is also used.[31]
12 The U(1) charge assignment is not unique.
A
8
wherewe assume γ0(X) = O((M / φ )7). If φ /√M Λ λ, thefermionmass
ij ℓ h i h i ℓ ∼
hierarchy (4) can be derived from the U(1) charge assignment:
A
Q (q¯ ) = (11,10,8) , Q (u ) = (4,2,0) , Q (d ) = (3,2,2) ,
A Li A Ri A Ri
¯
Q (l ) = (9,8,8) , Q (e ) = (6,4,2) , Q (h ) = 0 ,
A Li A Ri A (u)
Q (φ) = 1 , (25)
A
−
where we assume γ0(X) = O(Λ1/2M15/2/ φ 8). In either case, the quark flavor
ij ℓ h i
mixing due to the Kobayashi-Maskawa matrix (5) can be obtained by
1 λ λ3
(V ) λ|QA(q¯Li)−QA(q¯Lj)| λ 1 λ2 . (26)
KM ij  
∼ ∼
λ3 λ2 1
 
 
Finally, we discuss the neutrino sector. The Majorana mass matrix M
ν
of left-handed neutrinos is usually obtained throught the see-saw mechanism
such that[32, 33, 34]
v v
(M ) = f(ν) M−1 f(ν) , (27)
ν ij ia √2 R ab bj √2
(cid:16) (cid:17)
(ν)
where f is the Yukawa coupling among l , right-handed neutrinos ν
ia Li Ra
and h and (M ) is the superheavy Majorana mass matrix of right-handed
u R ab
neutrinos. In our Lifshitz type extension of the SM including Φ, M can
ν
be obtained without introducing right-handed neutrinos from the following
relevant interactions:
γ(ν)Φmi(jν)L¯ iτ τLc Htiτ τH , (28)
ij i 2 j · u 2 u
where L are fermions whose zero modes are l , and superscripts c and t
j Li
stand for the charge conjugation and transpose of the relevant field, respec-
(ν)
tively. The exponent m is determined by
ij
(ν) ¯ ¯
m Q (φ)+Q (l )+Q (l )+2Q (h ) = 0 . (29)
ij A A Li A Lj A u
In our model, M is given by
ν
γ0(ν) hφi m(ijν)v2 z (3+n−z)(m(ijν)+2) ,
(Mν)ij = γi0j(ν) MMℓℓ!2+n2−M2zℓ Mℓ 1+n2−z hφi m(ijνv)2 z <≥ (3+n−z)2(m(ijν)+2)! , (30)
 ij (cid:18) Λ (cid:19) (cid:18) Λ (cid:19) Λ  2Λ 2 !
9
0(ν)
where γ is a dimensionless parameter. Using a unitary matrix S , M is
ij ν ν
diagonarized as
S†M S = diag(m ,m ,m ) , (31)
ν ν ν ν1 ν2 ν3
where m < m < m . The lepton flavor mixing is given by the Maki-
ν1 ν2 ν3
Nakagawa-Sakata matrix V = S†S .[35] Using the experimental data for
MNS e ν
neutrinos,[18] we find that there are two large mixings:
sin22θ 0.88 , sin22θ > 0.92 (32)
12 23
∼
and the hierarchy between mass-squared differences for solar neutrinos ∆m2
⊙
and for the atmospheric neutrinos ∆m2:
⊕
∆m2 m2 m2
⊙ = | ν2 − ν1| λ2 . (33)
∆m2 m2 m2 ∼
⊕ | ν3 − ν2|
If three neutrino masses do not degenerate, the hierarchy (33) suggests the
relation:
(m ,m ) (λ,1) 0.05eV . (34)
ν2 ν3 ∼ ×
Using our mechanism with the U(1) charge assignment (24) or the ordinary
Froggatt-Nielsen mechanism with the U(1) charge assignment (25), M and
ν
V are estimated as
MNS
λ2 λ λ
(M ) λ∓(QA(¯lLi)+QA(¯lLj)) λ 1 1 , (35)
ν ij  
∝ ∝
λ 1 1
 
 
1 λ λ
V λ|QA(¯lLi)−QA(¯lLj)|S λ 1 1 S , (36)
MNS ν   ν
∼ ∼
λ 1 1
 
 
wheretheminusandtheplussignin(35)forourmechanismandtheFroggatt-
Nielsen mechanism, respectively. This type of neutrino mass matrix has been
proposed and studied in Refs. [36, 37, 38]. The above matrices (35) and (36)
haveaninteresting propertythatthebi-largemixing canbenaturallyderived
if we obtain the mass relation (34) after the diagonalization of M .
ν
Finally, we discuss the lepton number and/or baryon number violating
process through four fermi interactions. The four fermi interactions originate
10