Mirror World with Broken Mirror Parity, E(6) Unification and Cosmology

0.29 MB·English

by  C. R. Das

Mirror World with Broken Mirror
Parity, E Unification and Cosmology
6
C.R. Das 1 ∗, L.V. Laperashvili 2 †,
8
0 1 Center for High Energy Physics, Peking University, Beijing, China
0
2 2 The Institute of Theoretical and Experimental Physics, Moscow, Russia

n
a
J

7
2 Abstract

]
h In the present paper we have developed a concept of parallel ordinary (O) and
p mirror (M) worlds. We have shown that in the case of a broken mirror parity (MP),
-
p the evolutions of fine structure constants in the O- and M-worlds are not identical.
e It is assumed that E -unification inspired by superstringtheory restores the broken
6
h
MPatthescale∼ 1018 GeV,whatunavoidablyleadstothedifferentE -breakdowns
[ 6
2 at this scale: E6 → SO(10)×U(1)Z - in the O-world, and E6′ → SU(6)′ ×SU(2)′Z
- in the M-world. Considering only asymptotically free theories, we have presented
v
6 the running of all the inverse gauge constants α−1 in the one-loop approximation.
i
2
Then a ‘quintessence’ scenario suggested in Refs. [56–61] is discussed for our model
3
1 of accelerating universe. Such a scenario is related with an axion (‘acceleron’) of a
′
. new gauge group SU(2) which has a coupling constant g extremely growing at
2 Z Z
1 the scale ΛZ ∼ 10−3 eV.
7
0
:
v
i
X
r
a
∗ [email protected], [email protected]
† [email protected]
1 Introduction
In the present paper we consider the concept [1,2] (see also reviews [3,4]) that there exists
in Nature a ‘mirror’ (M) world – a hidden mirror sector – parallel to our ordinary (O)
world. The M-world as a mirror copy of the O-world contains the same particles and their
interactions as our visible world. Observable elementary particles of our O-world have
left-handed (V-A) weak interactions which violate P-parity. If a hidden mirror M-world
exists, then mirror particles participate in the right-handed (V+A) weak interactions and
have an opposite chirality. Lee and Yang were first [1] who suggested such a duplication of
the worlds which restores the left-right symmetry of Nature. The term ‘Mirror World’ was
introduced by Kobzarev, Okun andPomeranchuk inRef.[2], where they have investigated
a lot of phenomenological implications of such parallel worlds. The development of this
theory is given by Refs. [5–22].
The old idea of the existence of visible and mirror worlds became very attractive
over the last years in connection with a superstring theory [23–31]. Having a theory
′
described by the product G × G of symmetry groups corresponding to the parallel O-
and M-worlds, respectively, it is natural to associate it with superstring theory described
′
by E ×E [23,31].
8 8
Superstring theory is a paramount candidate for the unification of all fundamental
gauge interactions with gravity. Superstrings are free of gravitational and Yang-Mills
anomalies if a gauge group of symmetry is SO(32) or E ×E . The ‘heterotic’ superstring
8 8
′
theory E × E was suggested as a more realistic model for unification [23,31]. This
8 8
ten-dimensional Yang-Mills theory can undergo spontaneous compactification for which
E group is broken to E in four-dimensional space.
8 6
Among hundreds of papers devoted to the E -unification we should like to single
6
out Refs. [32–42].
If at small distances we have the E unification in our ordinary world, then we can
6
expect to have the same E unification in the mirror four-dimensional world assuming a
6
restoration of the left-right symmetry of Nature at small distances.
′
Wecanconsider aminimalsymmetryG ×G ,whereG = SU(3) ×SU(2) ×
SM SM SM C L
U(1) stands for the Standard Model (SM) of observable particles: three generations of
Y
′ ′ ′ ′
quarks and leptons and the Higgs boson, while G = SU(3) ×SU(2) ×U(1) is its
SM C L Y
mirror gauge counterpart having three generations of mirror quarks and leptons and the
mirror Higgs boson. The M-particles are singlets of G and O-particles are singlets of
SM
′
G . These different O- and M-worlds are coupled only by gravity (or maybe other very
SM
weak interaction).
′
In this paper all quantities of the mirror world will be marked by prime ().
′
A discrete symmetry ‘MP’ of the interchange G ↔ G is called ‘Mirror Parity’. If
this parity is conserved, then particle content of both sectors are identical and described
by the same Lagrangians with the same masses and coupling constants.
The aim of the present paper is to consider a case suggested in Refs. [43–45], when a
mirror parity MPis not conserved. Then, aswe show, theevolutions ofcoupling constants
in O- and M-worlds are different. The next assumption that mirror parity is restored by
theE -unificationatthescale∼ 1018 GeVinbothO-andM-worldsleadstothesignificant
6
consequences for cosmology.
1
In Section 2 we determine a mirror world and give particle contents existing in the
ordinary and mirror worlds.
Section 3 is devoted to symmetry groups considering only in the ordinary world. We
assume thatSM isextended by MSSM (Minimal Supersymmetric StandardModel). Then
its extension by left-right symmetry leads to the SO(10)-unification, which is ended by
the E -unification at the superGUT scale M ∼ 1018 GeV, according to the following
6 SGUT
chain:
SU(3) ×SU(2) ×U(1) → [SU(3) ×SU(2) ×U(1) ]
C L Y C L Y MSSM
→ SU(3) ×SU(2) ×SU(2) ×U(1) ×U(1)
C L R X Z
→ SU(4) ×SU(2) ×SU(2) ×U(1)
C L R Z
→ SO(10)×U(1) → E .
Z 6
All evolutions corresponding to these symmetry groups in the ordinary world are
presented in Figs. 1(a,b) and Figs. 2(a,b) for supersymmetry breaking scales 1 TeV and
10 TeV, respectively. Considering only asymptotically free symmetry groups, we have
used the one-loop approximations for these evolutions.
In Section 4 we have considered a mirror world with a broken mirror parity (MP).
We have shown the difference between evolutions of all fine structure constants in the O-
and M-worlds in the case of broken MP. To get the same E unification in both worlds,
6
′
we are forced to consider a quite different chain of the SM extension in the M-world. We
have assumed the following chain:
′ ′ ′ ′
[SU(3) ×SU(2) ×U(1) ] ×SU(2)
C L Y SM′ Z
′ ′ ′ ′
→ [SU(3) ×SU(2) ×SU(2) ×U(1) ]
C L Z Y SUSY
′ ′ ′ ′ ′
→ SU(3) ×SU(2) ×SU(2) ×U(1) ×U(1)
C L Z X Z
′ ′ ′ ′
→ SU(4) ×SU(2) ×SU(2) ×U(1)
C L Z Z
′ ′ ′
→ SU(6) ×SU(2) → E .
Z 6
All the evolutions corresponding to these symmetry groups in the mirror world are given
in Figs. 3(a,b) and Figs. 4(a,b) (in the one-loop approximation) for the supersymmetry
′
breaking scale M .
SUSY
A comparison of the evolutions considered in both worlds is shown in Figs. 5(a,b)
and Figs. 6(a,b).
All parameters of the evolutions are presented in Table 1.
′
A new mysterious gauge group SU(2) is considered in Section 5. We have cho-
Z
sen such a particle content of this group which leads to the ‘quintessence’ model of our
universe. The axion-like potential is investigated.
A new quintessence scenario in cosmology is developed in Section 6, together with
consequences for recent models of dark energy and dark matter. The problem of cosmo-
logical constant also is briefly discussed in this Section.
2
2 Particle content in the ordinary and mirror SM
In the Standard Model (SM) fermions are represented by Weyl spinors, however, the left-
handed (L) quarks and leptons: ψ = q ,l and right-handed (R) fermions: ψ = q ,l
L L L R R R
transform differently under SU(2)×U(1) symmetry. A global lepton charge is L = 1 for
leptons l ,l , and a baryon charge is B = 1 for quarks q ,q .
L R 3 L R
With the same rights we could formulate the SM in terms of antiparticle fields:
˜ ∗ ˜ ∗
ψ = Cγ ψ and ψ = Cγ ψ , where C is the charge conjugation matrix and γ is the
R 0 L L 0 R 0
Dirac matrix. These antiparticles have opposite gauge charges, opposite chirality, L = −1
for antileptons and B = −1 for antiquarks.
3
We can redefine the notion of particles considering L-particles and R-particles:
˜ ˜ ˜
L−fermions : ψ , ψ and R−fermions : ψ , ψ . (1)
L L R R
Including Higgs bosons φ we have the following SM content of the O-world:
L−set : (u,d,e,ν,u˜,d˜,˜e,N˜) ,φ ,φ ;
L u d
R˜ −set : (u˜,d˜,e˜,ν˜,u,d,e,N) ,φ˜ ,φ˜
R u d
˜ ∗
with φ = φ . (2)
u,d u,d
′
Considering the minimal symmetry G ×G we have the following particle content in
SM SM
the M-sector:
L′ −set : (u′,d′,e′,ν′,u˜′,d˜′,˜e′,N˜′) ,φ′ ,φ′ ;
L u d
R˜′ −set : (u˜′,d˜′,e˜′,ν˜′,u′,d′,e′,N′) ,φ˜′ ,φ˜′ . (3)
R u d
′
In general, we can consider a supersymmetric theory when G ×G contains grand
′ ′ ′
unification groups: SU(5)×SU(5), SO(10)×SO(10), E ×E , etc.
6 6
3 The SM and its extension in the ordinary world
In the present paper we consider the running of all the gauge coupling constants in
the SM and its extensions which is well described by the one-loop approximation of
the renormalization group equations (RGEs) from the Electroweak (EW) scale up to
the Planck scale. For simplicity of this investigation and with aim to demonstrate our
idea, we neglect the contributions of higher loops and Higgs bosons belonging to high
representations. Also we do not pay an attention to the realistic values of some unknown
scales (namely, a supersymmetric scale M , seesaw scale M , etc.) which we have
SUSY R
used in our numerical calculations. We give their reasonable values as examples.
For energy scale µ ≥ M , where M is the renormalization scale, we have the
ren ren
following evolution for the inverse fine structure constants α−1 given by RGE in the one-
i
loop approximation:
b
α−1(µ) = α−1(M )+ i t, (4)
i i ren 2π
where α = g2/4π, g are gauge coupling constants and t = ln(µ/M ).
i i i ren
3
3.1 Gauge coupling constant evolutions in the SM
We start with the SM in our ordinary world.
In the SM for energy scale µ ≥ M (here M is the top quark (pole) mass) we
t t
have the following evolutions (RGEs) [46–48] for the inverse fine structure constants α−1
i
(i = 1,2,3 correspond to the U(1), SU(2) and SU(3) groups of the SM), which are
L C
revised using updated experimental results [48] (see also Refs. [49,50]):
41
α−1(t) = 58.65±0.02− t, (5)
1 20π
19
α−1(t) = 29.95±0.02+ t, (6)
2 12π
7
α−1(t) = 9.17±0.20+ t. (7)
3 2π
Here M = M and t = ln(µ/M ). In Eq. (7) the value of α−1(M ) = 9.17 essentially
ren t t 3 t
depends on the value of α (M ) ≡ α (M ) = 0.118±0.002 (see [48]), where M is the
3 Z s Z Z
mass of Z-boson. The value of α−1(M ) is given by the running of α−1(µ) from M up
3 t 3 Z
to M , via the Higgs boson mass M . We have used the central value of the top quark
t H
mass M ≈ 174 GeV and M = 130±15 GeV.
t H
Evolutions (5)-(7) are shown from M up to the scale M in Fig. 1(a) and
t SUSY
Fig. 2(a), where x = log µ(GeV), t = xln10−lnM .
10 t
3.2 Running of gauge coupling constants in the MSSM
In this Subsection we consider the Minimal Supersymmetric Standard Model (MSSM)
which extends the conventional SM.
MSSM gives the evolutions (4) for α−1 (i = 1,2,3) from the supersymmetric scale
i
M (here M = M ) up to the seesaw scale M .
SUSY ren SUSY R
Figs. 1(a,b), 2(a,b) present also these evolutions which are given by the following
MSSM slopes b [46,47]:
i
33
b = − = −6.6, b = −1, b = 3. (8)
1 2 3
5
In Figs. 1(a,b), 2(a,b) we have presented examples with the scales M = 1 and 10
SUSY
TeV, respectively:
Figs. 1(a,b) are given for SUSY breaking scale M = 1 TeV and seesaw scale
SUSY
M = 1.25·1015 GeV; M ≈ 2.4·1017 GeV and α−1 ≈ 26.06.
R SGUT SGUT
Figs. 2(a,b) correspond to SUSY breaking scale M = 10 TeV and seesaw scale
SUSY
M = 2.5·1014 GeV; M ≈ 6.96·1017 GeV and α−1 ≈ 27.64.
R SGUT SGUT
3.3 Left-right symmetry as an extension of the MSSM
At the seesaw scale M the heavy right-handed neutrinos appear. We assume that the
R
following supersymmetric left-right symmetry [51–54] originates at the seesaw scale:
SU(3) ×SU(2) ×SU(2) ×U(1) ×U(1) . (9)
C L R X Z
4
Here we see additional groups SU(2)R and U(1)(B−L) originated at the scale MR. The
group U(1)(B−L) is mixed with the gauge group U(1)Y leading to the product U(1)X ×
U(1) of the two U(1) groups with quantum numbers X and Z linearly combined into
Z
the weak hypercharge Y [55]:
Y 1
= (X −Z). (10)
2 5
Considering the running (4) for the supersymmetric group (9) in the region µ ≥ M we
R
have M = M and the following slopes [46,47]:
ren R
b = b = −6.6, b = −9, b = 3. (11)
X 1 Z 3
Also the running of SU(2) ×SU(2) in the same region of µ is given by the slope:
L R
b = −2, (12)
22
and we have the following evolution:
1 µ
α−1(µ) = α−1(M )+ ln , (13)
22 22 R π M
R
where
α−1(M ) = α−1(M ). (14)
22 R 2 R
The next step is an assumption that the group SU(4) ×SU(2) ×SU(2) by Pati and
C L R
Salam [51] originates at the scale M giving the following extension of the group (9):
4
SU(3) ×SU(2) ×SU(2) ×U(1) ×U(1) → SU(4) ×SU(2) ×SU(2) ×U(1) . (15)
C L R X Z C L R Z
The scale M is given by the intersection of SU(3) with U(1) :
4 C X
α−1(M ) = α−1(M ). (16)
3 4 X 4
In the MSSM we have the following equation for the slope b of the SU(N) group (see
N
[46,47] and [41]):
1
b = 3N −N − N −N ·N −..., (17)
N f vector adjoint
2
where N is a number of flavors, N is a number of scalar Higgs fields in the fun-
f vector
damental representation and N is a number of Higgses in adjoint representation.
adjoint
Considering only the minimal content of scalar Higgs fields, e.g. quartets 4+¯4, we have
N = 2 and obtain from (17) the following slope for the running of α−1(µ):
vector 4
b = 3·4−6−1 = 5. (18)
4
Now the evolution (4) with M = M gives:
ren 4
5 µ
α−1(µ) = α−1(M )+ ln . (19)
4 4 4 2π M
4
This is the running for the symmetry group SU(4).
5
3.4 From SO(10) to the E -unification in the ordinary world
6
The intersection of α−1(µ) with the running of α−1(µ) leads to the scale M of the
4 22 GUT
SO(10)-unification:
SU(4) ×SU(2) ×SU(2) → SO(10), (20)
C L R
and we obtain the value of M from the relation:
GUT
α−1(M ) = α−1(M ). (21)
4 GUT 22 GUT
Then we deal with the running (4) for the SO(10) inverse gauge constant α−1(µ), which
10
runs from the scale M up to the scale M of the super-unification E :
GUT SGUT 6
SO(10)×U(1) → E . (22)
Z 6
The slope of this running is b .
10
In general, for the SO(N) group we have the following slope [41,46,47]:
3 1 1
SO(N)
b = (N −2)−N − N − (N −2)·N −.... (23)
N 2 f 2 vector 2 adjoint
Calculating the SO(10)-slope we must consider not only vectorial Higgs fields N = 2,
vector
but also N = 1, because the appearance of right-handed particles is impossible
adjoint
without adjoint Higgs field (see explanation in Ref. [50]). As a result, we obtain from
Eq. (23) the following SO(10)-slope:
b = 12−6−1−4 = 1. (24)
10
Then we have the following running of α−1(µ):
10
1 µ
α−1(µ) = α−1(M )+ ln , (25)
10 10 GUT 2π M
GUT
which is valid up to the superGUT scale M of the E -unification.
SGUT 6
Finally, as a result of our investigation, one can envision the following symmetry
breaking chain in the ordinary world:
E → SO(10)×U(1) → SU(4) ×SU(2) ×SU(2) ×U(1) →
6 Z C L R Z
SU(3) ×SU(2) ×SU(2) ×U(1) ×U(1) → SU(3) ×SU(2) ×U(1) .
C L R X Z C L Y
AllevolutionsofthecorrespondinginversefinestructureconstantsaregiveninFigs.1(a,b)
and 2(a,b).
4 Mirror world with broken mirror parity and mirror
scales
In this Section, as in Refs. [43–45] (see also [3,4]), our main assumption is the principle:
“The only good parity... is a broken parity”, what means that in general case the mirror
6
parity MP is not conserved in Nature. However, at the very small distances the mirror
′
parity is restored and super-unifications E and E (inspired by superstring theory) are
6 6
identical having the same M ∼ 1017 or ∼ 1018 GeV. By this reason, the superGUT
SGUT
scaleM maybefixedbytheintersection oftheevolutionsofgaugecoupling constants
SGUT
in both – mirror and ordinary – worlds, which were not identical from the beginning.
Now it is very interesting to discuss what particle physics exists in the mirror world
when the mirror parity MP is spontaneously broken.
If O- and M-sectors are described by the minimal SM with the Higgs doublets φ and
′
φ, respectively, then we can consider the Higgs potentials:
λ
U = −µ2φ+φ+ (φ+φ)2, (26)
4
and
′
λ
U′ = −µ′2φ′+φ′ + (φ′+φ′)2. (27)
4
′
In the case of non-conserved MP the VEVs of φ and φ are not equal:
′
2µ 2µ
′
v = 6= v = . (28)
λ λ′
′
FollowingRefs.[43–45],weassumethatv ≫ v andintroducetheparametercharacterizing
the violation of MP:
′
v
ζ = ≫ 1. (29)
v
As far as Yukawa couplings have the same values in both worlds, the masses of the SM
fermions and massive bosons in the mirror world are scaled up by the factor ζ:
′
m = ζm ,
q′,l′ q,l
′
M = ζM , (30)
W′,Z′,φ′ W,Z,φ
but photons and gluons remain massless in both worlds.
Let us consider now the following expressions:
b µ
α−1(µ) = i ln ,
i 2π Λ
i
— in the O-world, and
′
α′−1(µ) = bi ln µ (31)
i 2π Λ′
i
— in the M-world.
′
A big difference between v and v will not cause a big difference between scales Λ
i
′
and Λ (see [4,44]):
i
′
Λ = ξΛ with ξ > 1. (32)
i i
The values of ζ and ξ were estimated in Refs. [4,43–45]:
ζ ≈ 30 and ξ ≈ 1.5, (33)
7
as results of astrophysical implications of the mirror world with broken mirror parity. But
it is possible to have ζ in the region:
10 ≤ ζ ≤ 100. (34)
As for the neutrino masses, the same authors have shown that the theory with broken
mirror parity leads to the following relations:
m′ = ζ2m ,
ν ν
and
M′ = ζ2M , (35)
ν ν
where m are light left-handed and M are heavy right-handed neutrino masses in the O-
ν ν
′ ′
world, and m ,M are the corresponding neutrino masses in the M-world. These relations
ν ν
are valid for each of three generations.
4.1 Broken mirror parity and the running of gauge coupling
constants in the mirror SM
We assume that M-world is not P- and CP- invariant and differs with O-world.
′ ′ ′
Considering the mirror SM given by symmetry group G = SU(3) ×SU(2) ×
SM C L
′
U(1) , we deal with the following one-loop approximation RGE for the running of inverse
Y
fine structure constants α′−1(µ) (i = 1,2,3 correspond to the U(1)′, SU(2)′ and SU(3)′
i L C
groups of the mirror SM with broken MP):
′
α′−1(µ) = α′−1(M′ )+ bi t′, (36)
i i ren 2π
′ ′ ′
where M = ζM is the renormalization scale in the mirror world and t = ln(µ/M ).
ren ren ren
′
In the M-world we have scales Λ which are different with Λ (they are given by
i i
Eq. (32)), but O- and M-slopes are identical:
′
b = b . (37)
i i
Then in the SM of the M-sector we have the following evolutions:
′ −1 ′ −1 ′ bi ′ bi µ
(α) (µ) = (α) (M )+ t = ln , (38)
i i t 2π 2π Λ′
i
where
(α′)−1(M ) = α−1(M )− bi lnξ, (39)
i t i t 2π
or
(α′)−1(M′) = α−1(M ). (40)
i t i t
Finally, we obtain the following SM running of gauge coupling constants in the mirror
world:
8
(i)
′ −1 41 ′
(α) (µ) = 58.65±0.02− t , (41)
1 20π
(ii)
′ −1 19 ′
(α) (µ) = 29.95±0.02+ t , (42)
2 12π
(iii)
′ −1 7 ′
(α) (µ) = 9.17±0.20+ t , (43)
3 2π
′ ′
where t = ln(µ/M ). According to Eq. (30), the pole mass of the mirror top quark is
t
′
M = ζM .
t t
4.2 Mirror MSSM and a seesaw scale in the mirror world
If the Minimal Supersymmetric Standard Model (MSSM) extends the mirror SM in the
′
mirror world, then we can assume that mirror sparticle masses m˜ obey the relation
analogous to Eq. (30):
′
m˜ = ζm˜ . (44)
This relation leads to the assumption that the mirror SUSY-breaking scale is larger than
M :
SUSY
′
M = ζM . (45)
SUSY SUSY
The mirror MSSM gives the evolutions (36) for α′−1(µ) (i = 1,2,3) from the supersym-
i
′ ′ ′ ′
metric scale M (here M = M ) up to the GUT scale M .
SUSY ren SUSY GUT
Here it is worth the reader’s attention to observe that if heavy right-handed neutrino
′
masses are given by Eq. (35), then a mirror seesaw scale M obeys the following relation:
R
M′ = ζ2M . (46)
R R
According to the estimate (33) given by Refs. [4,43–45], we have:
M′ ∼ 103M . (47)
R R
NowifM ∼ 1014 GeV,thenM′ ∼ 1017 GeV,andaseesaw scaleisclosetothesuperGUT
R R
scale of the E -unification. This means that mirror heavy right-handed neutrinos appear
6
at the scale ∼ 1017 GeV.
Figs. 3(a), 4(a) present the mirror MSSM evolutions of α′−1(µ) (i = 1,2,3), where
i
slopes b are given by the same Eq. (8) as in the O-world [46,47]. In Figs. 3(a,b) we
i
′
have presented an example of the mirror MSSM evolution with the scale M = 10
SUSY
TeV, what corresponds to M = 1 TeV and ζ = 10. But in Figs. 4(a,b) we have
SUSY
′
shown an example of the mirror MSSM evolution with the scale M = 300 TeV, what
SUSY
corresponds to M = 10 TeV and ζ = 30.
SUSY
9