New Low-Energy Realization of the Superstring-Inspired E_6 Model

9 Pages·0.11 MB·English

by  Ernest Ma

UCRHEP-T155
January 1996
New Low-Energy Realization of
The Superstring-Inspired E Model
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Ernest Ma
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Department of Physics, University of California
n
a Riverside, California 92521, USA
J

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Abstract

1
v
8 The superstring-inspired E6 model is reduced to the supersymmetric standard
3
model extended by a specific U(1) factor. This choice allows for the existence of
2
1 naturally light singlet neutrinos which also mix with ν , ν , and ν , thus making it
e µ τ
0
6 possible to accommodate all present neutrino data. Other consequences of this model
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are also discussed: oblique corrections from Z-Z’ mixing, phenomenology of the two-
/
h
Higgs-doublet sector, and possible scenarios of gauge-coupling unification.
p
-
p
e
h
1 Introduction
:
v
i
X
r Thereareanumberofpossiblelow-energyrealizationsofthesuperstring-inspiredE6 model[1].
a
Their particle content is often that given by the fundamental 27 representation of E and the
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gauge group is often larger than that of the standard model by at least one U(1) factor. In
particular, there are two neutral singlets in each 27 representation, N and S, which may be
considered as “neutrinos” because they are very weakly interacting. The former (N) is the
Dirac partner of an ordinary doublet neutrino ν, whereas the latter (S) is not. A possible
scenario is that both N and S are heavy and that ν is light via the seesaw mechanism. In
that case, the neutrino sector is equivalent to that of the standard model with very small
Majorana neutrino masses. On the other hand, there are experimental indications at present
for three types of neutrino oscillations: solar[2], atmospheric[3], and laboratory[4], requiring
three mass differences which are not possible with only three neutrinos. Adding a fourth
1
doublet neutrino is not allowed because the invisible width of the Z boson tells us that there
should be only three such neutrinos. Hence the idea of one or more naturally light singlet
neutrinos should be entertained, and in this talk I will show how a properly chosen extra
U(1) factor allows for this possibility[5] and discuss also some of the other features of this
new model.
2 Reduction of the Superstring-Inspired E Model
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The fundamental 27 representation of E may be decomposed according to its maximum
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subgroup SU(3) SU(3) SU(3) :
C L R
× ×
27 = (3,3,1)+(3∗,1,3∗)+(1,3∗,3). (1)
Under the reductions SU(3) SU(2) U(1) and SU(3) U(1) U(1) , the
L → L × YL R → T3R × YR
individual left-handed fermionic components are defined as follows[6].
1 1 1 1 1
(u,d) (3;2, ;0,0), uc (3∗;0,0; , ), dc (3∗;0,0; , ), (2)
∼ 6 ∼ −2 −6 ∼ 2 −6
1 1 1 1 1 1 1 1
(ν ,e) (1;2, ;0, ), ec (1;0, ; , ), N (1;0, ; , ), (3)
e
∼ −6 −3 ∼ 3 2 6 ∼ 3 −2 6
1 1 1 1 1 1
(ν ,E) (1;2, ; , ), (Ec,Nc) (1;2, ; , ), (4)
E ∼ −6 −2 6 E ∼ −6 2 6
1 1 1 1
h (3;0, ;0,0), hc (3∗;0,0;0. ), S (1;0, ;0, ). (5)
∼ −3 ∼ 3 ∼ 3 −3
The electric charge is given here by
Q = T +Y +T +Y , (6)
3L L 3R R
and three families of the above fermions and their bosonic superpartners are assumed.
There are two possible SO(10) subgroups which also contain the SU(5) subgroup which
contains the standard SU(3) SU(2) U(1) gauge group. The conventional decompo-
C L Y
× ×
sition is
E SO(10) U(1) , (7)
6 → A × ψA
such that the 27 splits up as follows:
(16,1) = (u,d)+uc +ec +dc +(ν ,e)+N, (8)
A e
(10, 2) = h+(Ec,Nc)+hc +(ν ,E), (9)
− A E E
(1,4) = S. (10)
A
2
Note that all the usual quarks and leptons are contained in (16,1) , and the Higgs bosons
A
are in (10, 2) . The next step of the decomposition is
A
−
SO(10) SU(5) U(1) , (11)
A → × χA
such that
16 = (10,1)+(5∗, 3)+(1,5), 10 = (5, 2)+(5∗,2), 1 = (1,0). (12)
A A A
− −
Note that U(1) U(1) breaks down to U(1) via the adjoint 78, resulting in Q =
ψA × χA η η
T +5Y Q.
3L L
−
The alternative decomposition[7] is
E SO(10) U(1) , (13)
6 → B × ψB
with
(16,1) = (u,d)+uc +ec +hc +(ν ,E)+S, (14)
B E
(10, 2) = h+(Ec,Nc)+dc +(ν ,e), (15)
− B E e
(1,4) = N. (16)
B
The next step is then
SO(10) SU(5) U(1) , (17)
B → × χB
such that
16 = (10,1)+(5∗, 3)+(1,5), 10 = (5, 2)+(5∗,2), 1 = (1,0). (18)
B B B
− −
Note that S is trivial under U(1) , whereas N is trivial under U(1) .
χA χB
In the E superstring-inspired model, the Yukawa terms are supposed to be restricted
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to only those contained in 273 1, namely uc(uNc dEc), dc(uE dν ), ec(ν E eν ),
→ E − − E e − E
Shhc, S(EEc ν Nc), and N(ν Nc eEc). In the following, I also assume a Z discrete
− E E e E − 2
symmetry where all superfields are odd, except one copy each of (ν ,E), (Ec,Nc), and S,
E E
which are even. The bosonic components of the even superfields will serve as Higgs bosons
which break the electroweak gauge symmetry. Specifically, N˜c generates m , m , and m ;
h Ei u D 1
˜
ν˜ generates m , m , and m ; and S generates m and m . The mass matrix spanning
E d e 2 h E
h i h i
ν , N, ν , Nc, and S is then given by
e E E
0 m 0 0 0
D
 m 0 0 0 0 
D
M =  0 0 0 mE m1 . (19)
 
 0 0 mE 0 m2 
 
 
 0 0 m1 m2 0 
 
 
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3 Naturally Light Singlet Neutrinos
If the E superstring breaks only via the flux mechanism, then it is not possible to have only
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the standard-model gauge group. The latter must be extended by at least the U(1) factor
η
mentioned previously. Under U(1) , N and S transform identically [and so do (ν ,e) and
η e
(ν ,E) as well as dc and hc] but nontrivially, hence they are forbidden to have Majorana
E
mass terms N2 and S2. Therefore, in the above mass matrix , ν and N together make
e
M
up a Dirac neutrino of mass m , whereas m 2m m /m . If U(1) is actually broken at
D S 1 2 E η
∼
some large scale, then N and S may acquire large Majorana masses through gravitationally
induced nonrenormalizable interactions[8]. In that case, N and S are superheavy, whereas
m m2 /m .
ν ∼ D N
Toobtainnaturallylightsingletneutrinos, itisnowobviousthatU(1) shouldbereplaced
η
with U(1) ( U(1) ) under which
N ≡ χB
N 0, S 5, (u,d), uc, ec 1, dc, (ν ,e) 2, h, (Ec,Nc) 2, hc, (ν ,E) 3.
∼ ∼ ∼ e ∼ E ∼ − E ∼ −
(20)
Now it is possible to have both light doublet neutrinos: m m2 /m , and light singlet
ν ∼ D N
neutrinos: m 2m m /m . In addition, the ν Nc eEc mass term allows the two types
S ∼ 1 2 E e E −
of neutrinos to mix. Note that
Q = 6Y +T 9Y . (21)
N L 3R R
−
4 Oblique Corrections from Z-Z’ Mixing
The proposed symmetry breaking is achieved through a combination of the adjoint 78 which
breaks E down to SU(3) SU(2) U(1) U(1) U(1) and a pair of superheavy
6 C × L× YL × T3R × YR
27 and 27∗ which break it down to SO(10) , resulting in the intermediate gauge group
B
G = SU(3) SU(2) U(1) U(1) . (22)
C L Y N
× × ×
Then U(1) is broken by S˜ u, and SU(2) U(1) breaks down to U(1) via ν˜ v ,
N L Y Q E 1
h i ≡ × h i ≡
and N˜c v . Let
h Ei ≡ 2
v
2 2 2 2 2 2 2
v = v +v , tanβ = , g = g +g , (23)
1 2 v Z 1 2
1
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then the observed Z boson is a linear combination of Z from SU(2) U(1) and Z from
1 L Y 2
×
U(1) : Z = Z cosθ+Z sinθ, where
N 1 2
1 3 2 v2 2 g 3 v2
2 2 2 2 Z 2
M g v 1 sin β , θ sin β . (24)
Z ≃ 2 Z " −(cid:18) − 5(cid:19) u2# ≃ −s5 gN (cid:18) − 5(cid:19) u2
The deviations from the standard model may be expressed in terms of the conventional
oblique parameters:
9 v2
4
ǫ = αT = sin β , (25)
1
− 25 u2
(cid:18) (cid:19)
αU 3 v2
2
ǫ = = sin β , (26)
2 −4sin2θ − 5 u2
W (cid:18) (cid:19)
αS 2 1 3 v2
2
ǫ = = 1+ sin β . (27)
3 4sin2θ 5 4sin2θ − 5 u2
W (cid:18) W(cid:19)(cid:18) (cid:19)
The present precision data from LEP at CERN have errors of order a few 10−3. This
×
means that u TeV is allowed.
∼
5 Phenomenology of the Two-Higgs-Doublet Sector
At the electroweak energy scale, there are only two Higgs doublets in this model, but they
differ[9] from the ones of the minimal supersymmetric standard model (MSSM). The reason
is that the superpotential has the term f(ν Nc EEc)S which has no analog in the MSSM.
E E−
Let
φ¯0 ν˜ φ+ E˜c
Φ˜ 1 E , Φ 2 , χ S˜, (28)
1 ≡  φ−  ≡  E˜  2 ≡  φ0  ≡  N˜c  ≡
− 1 2 E
       
then the quartic terms of the Higgs potential are given by the sum of
V = f2[(Φ†Φ )(Φ†Φ )+(Φ†Φ +Φ†Φ )(χ¯χ)], (29)
F 1 2 2 1 1 1 2 2
and
1
V = g2[(Φ†Φ )2 +(Φ†Φ )2 +2(Φ†Φ )(Φ†Φ ) 4(Φ†Φ )(Φ†Φ )]
D 2 1 1 2 2 1 1 2 2 1 2 2 1
8 −
1
+ g2[(Φ†Φ )2 +(Φ†Φ )2 2(Φ†Φ )(Φ†Φ )]
1 1 1 2 2 1 1 2 2
8 −
1
+ g2 [9(Φ†Φ )2 +4(Φ†Φ )2 +12(Φ†Φ )(Φ†Φ ) 30(Φ†Φ )(χ¯χ)
80 N 1 1 2 2 1 1 2 2 − 1 1
20(Φ†Φ )(χ¯χ)+25(χ¯χ)2]. (30)
2 2
−
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Let χ = u, then √2Reχ is a physical scalar boson with m2 = (5/4)g2 u2, and couples to
h i N
Φ†Φ with strength √2u(f2 (3/8)g2 ). The effective (Φ†Φ )2 coupling λ is then given by
1 1 − N 1 1 1
1 9 2(f2 (3/8)g2 )2
λ = (g2 +g2)+ g2 − N . (31)
1 4 1 2 40 N − (5/4)g2
N
The other quartic self-couplings of the reduced Higgs potential involving only Φ and Φ
1 2
have similar additional contributions. Consequently,
1 6 8f4
2 2 2
λ = (g +g )+ f , (32)
1 4 1 2 5 − 5g2
N
1 4 8f4
2 2 2
λ = (g +g )+ f , (33)
2 4 1 2 5 − 5g2
N
1 1 8f4
2 2 2
λ = g + g +f , (34)
3 −4 1 4 2 − 5g2
N
1
2 2
λ = g +f . (35)
4 2
−2
The MSSM is recovered in the limit of f = 0 as expected. Otherwise, the two-Higgs-doublet
structure is different. In particular,
1 3 1 1 8f4
2 2 2 2 2 2
(m ) = 2v g cos 2β +f + cos2β cos 2β +ǫ, (36)
h max "4 Z (cid:18)2 5 − 2 (cid:19)− 5gN2 #
where ǫ comes from radiative corrections, the largest contribution being that of the top
quark:
3g2m4 m˜2
ǫ 2 t ln 1+ . (37)
≃ 8π2MW2 m2t!
Normalizing g2 to be equal to (5/3)g2 and varying f2,
N 1
25 3 1 1 2
2 2 2 2 2
(m ) < M cos 2β + sin θ + cos2β cos 2β +ǫ. (38)
h max Z " 24 W (cid:18)2 5 − 2 (cid:19) #
Numerically, the maximum value occurs at β = 0, which corresponds to m 140 GeV, as
h
≃
compared to 128 GeV in the MSSM.
6 Possible Scenarios of Gauge-Coupling Unification
Consider now the issue of gauge-coupling unification. The evolution equations of α g2/4π
i ≡ i
are generically given to two-loop order by
∂α 1 b
i ij 2
µ = b + α (µ) α (µ), (39)
∂µ 2π " i 4π j # i
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where µ is the running energy scale and the coefficients b and b are determined by the
i ij
particle content of the model. To one loop, the above equation is easily solved:
b M
α−1(M ) = α−1(M ) i ln 1. (40)
i 1 i 2 − 2π M
2
Below M , assume the standard model with 2 Higgs doublets, then
SUSY
21
b = , b = 3, b = 7. (41)
1 2 3
5 − −
Above M in the MSSM,
SUSY
3 1 1
b = 3(2)+ (4) , b = 6+3(2)+2 , b = 9+3(2). (42)
1 2 3
5 4 − 2 −
(cid:18) (cid:19) (cid:18) (cid:19)
It is now well-known that for M 104 GeV, the gauge couplings do in fact unify at
SUSY
∼
M 1016 GeV.
U
∼
In the present model with u M ,
SUSY
∼
b = 3(3) + ?, b = 6+3(3) + ?, b = 9+3(3) + ?, b = 3(3) + ? (43)
1 2 3 N
− −
There are two possible scenarios for gauge-coupling unification. The first is an analog of the
MSSM. Add one extra copy of (ν ,e) and (Ec,Nc) [having Q = 0 so that the theory
e E N
remains anomaly-free], then
P
3 2
∆b = , ∆b = 1, ∆b = 0, ∆b = . (44)
1 2 3 N
5 5
This again leads to M 1016 GeV, which is actually not so good because the string scale
U
∼
is an order of magnitude higher. Also, it is hard to understand theoretically why the chosen
superfields are light but their companions in the same E multiplet are superheavy. On the
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other hand, this is no worse than the usual assumptions taken in SUSY SU(5) or SUSY
SO(10).
The second scenario is to insist on having M 5 1017 GeV, and allow some compo-
U
∼ ×
nents of the superheavy 27 and 27∗ multiplets to be somewhat lighter than the others. In
particular, take 3 copies of (u,d)+ (u∗,d∗) and (ν ,e)+ (ν∗,e∗) with M′ much below M ,
e e U
then between M′ and M ,
U
1 3 12
∆b = 3 + = , ∆b = 3 (3+1) = 12, ∆b = 3 (2+0) = 6, (45)
1 2 3
× 5 5 5 × ×
(cid:18) (cid:19)
3 2 21
∆b = 3 + = . (46)
N
× 10 5 10
(cid:18) (cid:19)
As a result, gauge-coupling unification at the string scale is achieved with an intermediate
scale of M′ 1016 GeV.
∼
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7 Conclusions
In conclusion, the superstring-inspired E model provides a framework for accommodating
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naturally light singlet neutrinos as well as naturally light doublet neutrinos which also mix
with each other. The key is the reduction
E SU(3) SU(2) U(1) U(1) , (47)
6 C L Y N
→ × × ×
under which N is trivial, but S is not. Hence N may acquire a large Majorana mass m ,
N
and the mass matrix of Eq. (19) becomes
M
0 m 0 m 0
D 3
 m m 0 0 0 
D N
M =  0 0 0 mE m1 . (48)
 
 m3 0 mE 0 m2 
 
 
 0 0 m1 m2 0 
 
 
This means that the doublet neutrinos obtain very small masses from the usual see-saw
mechanism: m m2 /m , whereas the singlet neutrinos S get theirs from an analogous
ν ∼ D N
3 3 mass matrix: m 2m m /m .
S 1 2 E
× ∼
Other properties of this model include: (1) the two-Higgs-doublet structure at the elec-
troweak energy scale is not that of the minimal supersymmetric standard model; (2) an
additional neutral gauge boson (Z’) is possible at the TeV energy scale; and (3) gauge cou-
plings may unify at the string compactification scale if there can be variations of masses
within some superheavy supermultiplets.
Acknowledgments
I thank Profs. G. Zoupanos and N. Tracas, and all the other organizers of the 5th Hellenic
School and Workshops on Elementary Particle Physics for their great hospitality and a very
stimulating program. This work was supported in part by the U. S. Department of Energy
under Grant No. DE-FG03-94ER40837.
References
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