A symmetry breaking mechanism by parity assignment in the noncommutative Higgs model

0.12 MB·English

by  Masaki J. S. Yang

KEK-TH-1791
A symmetry-breaking mechanism by parity assignment
5
1 in the noncommutative Higgs model
0
2

v
o
Masaki J.S. Yang
N

1 Institute of Particle and Nuclear Studies,

High Energy Accelerator Research Organization (KEK),
]
h Tsukuba 305-0801, Japan
p
-
p Abstract
e
h
Weapplytheorbifoldgrandunifiedtheory(GUT)mechanismtothenoncommu-
[
tative Higgs model. An assignment of Z parity to the “constituent fields” induces
2
2
parity assignments of both the gauge and Higgs bosons, because these bosons are
v
8 treated as some kind of composite field in this formalism. As a result, some of the
8 gauge bosons and the colored triplet Higgs boson receive heavy mass comparable to
8
the GUT scale, and the gauge symmetry is broken. No particles appear other than
3
0 the SM ones in the massless states.
.
1
0
5
1
:
v
i
X
r
a
1 Introduction
The discovery of the Higgs boson [1,2], a great triumph of science, has completed the
particle contents of the Standard Model (SM). However, the theoretical origin of the
Higgs boson is still unclear. Innumerable theories and models have been considered so
far, in order to explain the existence of the boson and clarify the underlying physics
beyond the SM.
Among these attempts, one interesting possibility is the noncommutative geometry
inspired by the Yang–Mills–Higgs model, developed by A. Connes [3,4]. We call this
model noncommutative Higgs model (NHM) for short. In this picture, the underlying
spacetime is assumed to be a multi-sheet of ordinary Minkowski space M4 Z . The
n
×
Higgs boson is interpreted as a gauge boson along a discrete extra dimension, which has
noncommutative differential algebra. For example, in the simplest model M4 Z , the
2
×
coordinates are represented as (xµ,y = 1). Thus, y2 = 1 holds, and it leads to the anti-
±
commutative relation ydy = dyy. This noncommutative differential algebra generates
−
nonzero Higgs mass and the spontaneous symmetry breaking (SSB) mechanism.
This concept is widely followed by its successors [5–8] and explained in several reviews
[9–12]. It is also applied to supersymmetry [13], and the grand unified theories (GUT),
e.g., SU(5) [14–17] or SO(10) models [18,19]. Meanwhile, when this kind of model is
regarded as a theory with an extra dimension, several mechanisms and structures in the
normal extra dimension can be recast to the noncommutative extra dimension.
As a typical example, in this paper, we implement the orbifold GUT mechanism by
Y. Kawamura [20] in the SU(5) NHM. In the previous standard studies of the NHM, the
symmetry-breaking pattern is determined by selecting the distance matrices M . By
nm
contrast, we imposed an assignment of Z parity to the “constituent fields”. This induces
2
parity assignments of both gauge and Higgs bosons, because these bosons are treated
as some kind of composite field in this formalism. This is the main difference from the
original orbifold GUT model, which treats the assignments of bosons as independent
conditions. As a result, some of the gauge bosons and the colored triplet Higgs boson
receive heavy mass comparable to the GUT scale, and the SU(5) gauge symmetry is
broken. No particles appear other than the SM ones in the massless states.
This paper is organized as follows. In the next section, we review the generalized
gauge theory in the NHM. In Sect. 3, we present the symmetry-breaking mechanism by
parity assignment, and discuss a new field and proton decay. Section 4 is devoted to the
conclusion.
2 Generalized gauge theory on M4 Z
n
×
Here, wedefineseveral definitionsandthemainframeoftheformulationinthegeneralized
gaugetheoryonM4 Z . SincetheformulationisthesameasthoseinRef.[16], adetailed
n
×
description is omitted. The essential difference in the formulations between the original
paper [14,15] and Ref. [16] is the flavor dependence of the distance matrices M ; the
nm
differential algebraic structure is almost equivalent.
1
2.1 Differential calculus
On the n-sheeted Minkowski space M4 Z , the coordinates are represented by (xµ,y =
n
×
1 N), and a function on the nth sheet is expressed as f(x,n) f . In this space, the
n
− ≡
exterior derivative is generalized to d = d+d , introducing extra exterior derivatives d
χ χ
and the differential forms χ . They are defined as follows:
m
df = (d+d )f , df = ∂ f dxµ,
n χ n n µ n
d f = d f = [M f f M ]χ ,
χ χm nm m − n nm m
mX6=n mX6=n
where the matrices M† = M (n = m) represent the distances between two sheets. In
nm mn 6
order to keep the nilpotency of the generalized exterior derivative d2 = 0, the differential
forms dxµ and χ should be anti-commutative, dxµ χ = χ dxµ , and the d should
m m n χ
∧ − ∧
also be nilpotent, d2 = 0. To prove d2 = 0, and to keep the differential algebra consistent,
χ χ
there are several “index shifting rules” between f ,M , and χ . These shifting rules are
n nm m
the source of the noncommutativity that corresponds to the relationydy = dyy in other
−
formulations [5–8]. The precise form of these rules and a proof of d2 = 0 is presented in
χ
Ref. [16].
2.2 Generalized connection
In several classes of models [5–8], the gauge and Higgs bosons are regarded as elemental
fields. By contrast, Connes’s original paper and its successors [3,14,15] treat these bosons
assomekindofcompositefield. Thisformulationiseffective inconstructing realisticGUT
modelsinparticular. Here we adoptthiscomposite formulation, which isalso quotedfrom
Ref. [16]. In this picture, a gauge field consists of “constituent fields”, defined on the same
sheet, while a Higgs field is defined on the different sheets. The quotation marks “ ” mean
that the detailed dynamics of the binding mechanism is not specified.
The above picture is described by utilizing the generalized connection one-form A,
defined as
A(x,n) a†(x,n)da (x,n),
≡ i i
Xi
= a†(x,n)da (x,n)+ a†(x,n)[M a (x,m) a (x,n)M ]χ ,
i i i nm i − i nm m
Xi Xi mX6=n
A(x,n)+ Φ (x)χ . (1)
nm m
≡
X
m6=n
Here, the“constituentfield”a (x,n)isasquare-matrix-valuedcontinuousfunctiondefined
i
on the nth sheet, and the summation over i is assumed to be a finite sum. The last line
2
in Eq. (1) defines the gauge and Higgs fields as follows:
A(x,n) = a†da (x,n), (2)
i i
Xi
Φ (x) = a†(x,n)[M a (x,m) a (x,n)M ] (n = m). (3)
nm i nm i − i nm 6
Xi
Henceforth, we use the notation A(x,n) = A ,A(x,n) = A , and a (x,n) = ai as
n n i n
shortened forms. As in Refs. [14–16], we impose the following Hermitian condition
ai†ai = 1, A† = A , ( Φ χ )† = Φ χ . (4)
n n ⇒ n − n nm m − mn n
Xi mX6=n mX6=n
From Eq. (4), we assume that Φ† = Φ and χ† = χ . For later convenience, the
nm mn m − m
back-shifted Higgs field
H Φ +M = ai†M ai , (5)
nm ≡ nm nm n nm m
Xi
is also introduced.
The field-strength two-form is defined as follows:
F = dA +A A , (6)
n n n n
∧
where
dA = dai† dai, (7)
n n ∧ n
Xi
because of d2 = 0. After some calculation, the explicit form of F contains the following
n
three elements:
F = F dxµ dxν + D H dxµ χ + (H H X′ )χ χ , (8)
n nµν ∧ µ nm ∧ m nm ml − nml m ∧ l
mX6=n m6=Xn,l6=m
where
1
F (x,n) F = (∂ A (x,n) ∂ A (x,n)+[A (x,n),A (x,n)]) (9)
µν nµν µ ν ν µ µ µ
≡ 2 −
D H (x) = ∂ H (x)+A (x,n)H (x) H (x)A (x,m), (10)
µ nm µ nm µ nm nm µ
−
X′ (x) = a†(x,n)M M a (x,l). (11)
nml i nm ml i
Xi
It contains the gauge boson, the Higgs boson, and the new field X′ (x). The treatment
nml
of X′ is decided by whether X′ is a dependent function of the Higgs fields H or
nml nml nm
not. If X′ can be written as some function of the Higgs fields X′ = f(H ), X′
nml nml nm nml
is treated as a Higgs interaction terms. If not, X′ is regarded as an auxiliary field that
nml
does not have a kinetic term, and then it will be eliminated from the Lagrangian by the
equation of motion ∂ /∂X′ = 0.
L nml
3
In order to determine the gauge transformation of “composite fields”, the “constituent
field” ai is first assigned to be a fundamental representation under the nth gauge trans-
n
formation:
ai′ = aig , (12)
n n n
where g = g(x,n) = (g(x,n)†)−1 is anarbitraryunitary matrix associated with thegauge
n
group on the sheet n. From Eq. (12), the generalized connection and field strength will
transform as the standard form:
A′ = g−1A g +g−1dg , (13)
n n n n n n
F′ = g−1F g , (14)
n n n n
′
with M = M . In addition, Eq. (13) implies that the back-shifted Higgs field (5)
nm nm
transforms as a bifundamental representation:
H′ = g−1H g . (15)
nm n nm m
2.3 The Lagrangian
In consequence, the gauge-invariant Lagrangian is formulated by
1 1
= tr F ,F , (16)
LYMH −4 g2 h n ni
Xn n
whereindependent coupling constantsg areintroducedforgaugefields oneachnthsheet.
n
In order to calculate Eq. (16), the metric of the space M4 Z is specified as
n
×
dxµ,dxν = gµν,
h i
χ ,dxµ = 0,
n
h i
χ ,χ = δ α2,
h n mi − nm n
where gµν = diag(+, , , ). Then the inner products of the two-forms are found to be
− − −
dxµ dxν,dxρ dxσ = gµρgνσ gµσgνρ, (17)
h ∧ ∧ i −
dxµ χ ,dxν χ = δ α2gµν, (18)
h ∧ m ∧ ni − nm n
χ χ ,χ χ = α2α2 (δ δ δ δ ), (19)
h k ∧ l n ∧ mi n m kn lm − km ln
while the other inner products among the basis two-forms are found to be zero. Finally,
we split X (x) in Eq. (11) into two terms according to n = l and n = l for convenience:
nml
6
P (x) = H (x)H (x) X′ (x), (20)
nm nm mn − nmn
Q (x) = H (x)H (x) X′ (x), (21)
nml nm ml − nml
4
where n = m = l = n. From Eq. (16), we obtain the final expression of the (bosonic
6 6 6
sector of the) Lagrangian [16]:
1 α2
= tr F (x,n) 2 + mtr D H (x) 2
LYMH − 8g2 | µν | 4g2 | µ nm |
Xn n Xn nX6=m n
α2 α2 α2 α2
m ntr P (x) 2 m l tr Q (x) 2. (22)
nm nml
− 4g2 | | − 4g2 | |
Xn nX6=m n Xn m6=Xn,l6=n n
TheLagrangian(22)issubdivided intofourtermscorrespondingtothedecomposition(8):
The first term is the pure Yang–Mills term with independent coupling constants, the
second is the Higgs kinetic energy term, the third represents the self-coupling of Higgs
H , and the fourth term describes interactions among different Higgs H and H .
nm nm ml
3 SU(5) GUT-breaking by parity assignment
In this section, we review an SU(5) GUT in the NHM briefly, and implement the orbifold
GUT mechanism in this toy model. In the formalism presented in the previous section,
an SU(5) GUT model requires N ≧ 3. This is because N ≧ 3 realizes more than two
independent M , which correspond to two energy scales of SU(5) symmetry breaking,
nm
SU(5) SU(3) SU(2) U(1) and SU(3) SU(2) U(1) SU(3) U(1) .
c L Y c L Y c em
→ × × × × → ×
Then we choose N = 3 to describe the SU(5) GUT [14,16]. The indices n,m,l in Eq. (22)
then take the values 1, 2, 3 only.
Aa
1µ
H
♠⑤ 1
Σ ♠⑤
H
♠⑤ 2
Aa
2µ
Figure 1: A schematic image of the SU(5) model on M4 Z .
3
×
At first, the “constituent fields” ai,ai are assumed to be complex 5 5 matrices and
1 2 ×
ai has a real-valued continuous function that satisfy Eq. (4), ai†ai = 1. Additionally,
3 i n n
we consider a parity symmetry between n = 1 and n = 2 sheetPs of spacetime, and impose
the following parity condition for the “constituent fields”:
ai = PaiP, (23)
2 1
where P = diag ( 1, 1, 1,+1,+1). In order to break the gauge symmetry, this parity
− − −
assignment is found to be unique under proper assumptions, which are discussed later.
5
The SU(5) gauge fields at each of the three copies of M4 are calculated as follows:
A(x,1) = ai†dai = iTaAa(x) A, (24)
1 1 1 ≡
Xi
A(x,2) = ai†dai = iTaAa(x) PAP, (25)
2 2 2 ≡
Xi
A(x,3) = ai†dai = 0, (26)
3 3
Xi
where Ta(a = 1, ,24) are the generators of SU(5), provided that the following traceless
···
condition is imposed:
TrA(x,1) = TrA(x,2) = 0. (27)
In order to determine Higgs fields, we set the matrix M as
nm
M = M = Mdiag(1,1,1,1,1) Σ , (28)
12 21 0
≡
M = M = M† = M† = µ 0 0 0 0 1 T H , (29)
13 23 31 32 ≡ 0
(cid:0) (cid:1)
where M(µ) is the energy scale at the stage of GUT (SM) symmetry breaking SU(5)
→
SM (SM SU(3) U(1) ). Thus, the back-shifted Higgs fields are found to be
c em
→ ×
Σ(x)+Σ = H (x) = ai†MPaiP = PH (x)P, (30)
0 12 1 1 21
Xi
H(x)+H = H (x) = ai†M ai = PH (x). (31)
0 13 1 13 3 23
Xi
From Eqs. (30) and (31), it is found that H is a 5 1 matrix-valued field transforming
13
×
like 5 representation under SU(5), and H is a 5 5 matrix-valued field that is a linear
12
×
combination of the field like 1 and 24 representation. The discussion on the field Σ(x) is
presented later. Note that both the parity assignments of the gauge boson (25) and the
Higgs boson (31) are determined by the condition of the “constituent fields” (23). They
are independent conditions in the original orbifold GUT model.
In order to calculate the Lagrangian (22), it is necessary to consider which X′ (x)
nml
are independent of the Higgs field and which X′ (x) are not. Here we refer to only
nml
dependent X′ (x), which should be kept in the Lagrangian:
nml
X′ = X′ = ai†M M ai = M2, (32)
121 212 1 12 21 1
Xi
X′ = X′ = ai†M M ai = µ2, (33)
313 323 3 31 13 3
Xi
X′ = PX′ = ai†M M ai = M(H +H ), (34)
123 213 1 12 13 3 1 0
Xi
X′ = X′ P = ai†M M ai = M(H +H )†P. (35)
312 321 3 31 12 2 1 0
Xi
6
Since other X′ (x) are auxiliary fields independent of the Higgs, the corresponding
nml
P (x) and Q (x) are entirely eliminated in Eq. (22) by ∂ /∂X′ = 0. Substitut-
nm nml L nml
ing these results into Eq. (22), we find the final form of the Lagrangian:
1 α2 α2 β2
= trF† Fµν + tr D (Σ+Σ ) 2 + + D (H +H ) 2
LSU(5) − 4g2 µν 2g2 | µ 0 | (cid:18)2g′2 2g2(cid:19)| µ 0 |
α4 α2β2
tr Σ+Σ 2 M2 2 H +H 2 µ2 2
− 2g2 || 0| − | − 2g′2 || 0| − |
α2β2 α4
+ (ΣP +MP M)(H +H ) 2. (36)
−(cid:18) 2g2 2g′2(cid:19)| − 0 |
Here, F = ∂ A ∂ A + [A ,A ], and we have assumed the 1 2 symmetry g2 =
g2 = g2,µαν 2 = αµ2 =ν −α2,νanµd set gµ2 =νg′2,α2 = β2. ↔ 1
2 1 2 3 3
In particular, the mass term of the 5 representation Higgs is computed as
(MP M)H 2 = M2diag(4,4,4,0,0)H†H. (37)
| − |
Similarly, the gauge boson masses are
D H 2 (A M MPA P)2 = (MAaˆTaˆ)2, (38)
| µ 12| ∋ µ − µ µ
where aˆ runs the broken generators except those of SU(3) SU(2) U(1) . Equa-
c L Y
× ×
tions (37) and (38) show that the parity assignment condition (23), ai = PaiP, invokes
2 1
SU(5)symmetry breaking, andprovidesthecoloredtripletHiggsandbrokengaugebosons
with heavy mass of order M. Therefore, it is reasonable to consider that the symmetry
breaking by the condition (23) corresponds to the orbifold GUT mechanism [20] in the
NHM. This is our main result. Of course, this result holds only at tree-level. However,
if the parity symmetry imposed between the first and second sheets is not broken by the
quantum correction, mixing between the Higgs doublet and triplet is prohibited at the
quantum level.
3.1 Discussions
By imposing the condition (23), the additional Higgs field Σ(x) emerges (in the case of
P = 1, H (x) = aiMai = M and then Σ(x) disappears ). To investigate whether it
12 i 1 1
is possible to elimPinate the Σ field or not, let us consider the most general form of the
parity assignment:
a = Oa P, a = Qa R. (39)
i1 i i2 i
Here we assume that O,P,Q,R are all diagonal and commutative, and O2 = P2 = Q2 =
R2 = 1, which fulfills the Hermitian condition (4). OPQR = 1 is also imposed so as
to give Σ(x) 0 at the proper gauge transformation. In this case, gauge symmetry
→
breaking in the Lagrangian (36) requires the conditions O = Q and P = R. Since only
6 6
7
the differences between them are important, we can set O = P = 1. Then Q = R holds by
OPQR = 1, which leads to the condition (23). On the other hand, in order to eliminate
the Σ field, O = Q should hold. Then, these two conditions, gauge symmetry breaking
and elimination of the Σ field, are incompatible. Under these proper assumptions, we
conclude that it is impossible to avoid this kind of field to implement the orbifold GUT
mechanism in this formalism.
In order to probe the gauge transformation property of this Σ field, the parity matrix
P is decomposed into a linear combination of the hypercharge Y and identity matrix I
as follows:
1
Y = diag(2,2,2, 3, 3), (40)
√60 − −
2 1
P = diag( 1, 1, 1,1,1) = √60Y I cY +dI. (41)
− − − −5 − 5 ≡
Then, the Higgs field is found to be
H P = ai†MPai = M ai†(cY +dI)ai (42)
12 1 1 1 1
Xi Xi
= cM ai†Yai +dM. (43)
1 1
Xi
Since Y is the generator of SU(5), the first term in Eq. (43) behaves as an adjoint Higgs
field.
Finally, we will mention the nucleon stability. In this toy SU(5) model, heavy gauge
bosons of broken symmetry X ,Y and colored triplet Higgs Hc induce nucleon decay.
µ µ
However, thisisanintrinsicprobleminSU(5)andwecanextendthelifetimeofthenucleon
by other mechanisms, e.g., supersymmetrization. However, couplings between fermions
and X ,Y ,Hc bosons can be forbidden by the proper parity assignment of fermions in
µ µ
the normal orbifold GUT [21]. There is, thus, a possibility that these couplings could also
be prohibited in this formalism. However, the fermionic sector in the NHM has subtleties
and there are several definitions [8,12,14]. We leave the construction of the fermionic
sector for future work.
4 Conclusions
In this paper, we have implemented the orbifold GUT mechanism in the noncommutative
Higgs model. An assignment of Z parity to the “constituent fields” induces parity as-
2
signments of both the gauge and Higgs bosons, because these bosons are treated as some
kind of composite field in this formalism. This is the main difference from the original
orbifold GUT model, which treats the assignments of bosons as independent conditions.
As a result, some of the gauge bosons and the colored triplet Higgs boson receive heavy
mass comparable to the GUT scale, and the SU(5) gauge symmetry is broken. No par-
ticles appear other than the SM ones in the massless states. In fact, nucleon decay is a
8
problem in this model. However, there is a possibility that couplings between fermions
and X ,Y ,Hc bosons could be forbidden by the proper parity assignment of fermions.
µ µ
We leave the construction of the fermionic sector for future work.
Acknowledgements
The author would like to acknowledge R. Kitano for useful discussions and valuable com-
ments. This study is supported by JSPS Research Fellowships for Young Scientists, No.
24 8357.
·
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