Bosonic SeeSaw Mechanism

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Review of heterotic string constructions
J. E. Kim
Seoul National Univ.
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• 1. Introduction
• 2. Orbifold geometry
• 3. Application in string theory
• 4. Phenomenology
• 5. Z12 example

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1. Introduction
The SM has been constructed as a series of clever
observations since the discovery of parity
violation. It is a chiral theory, L-R
non-symmetric fermion representations at least at
the electroweak scale. In this sense, chirality
of the SM is the heart of physical laws at low
energy.
There have been many ideas, V-A theory, quark
mixing, current algebra, spontaneous symmetry
breaking, introduction of quarks and color
degrees, asymptotic freedom, instanton solutions,
etc. But the appearance of chirality is the most
fundamental thing to have the low energy world as
we observe today.
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The outstanding problems of the SM are

• 15(1) chiral fermions in one family
• Number families ? 3, probably exactly 3
• N1 supersymmetry and achievement of gauge hier.
• Doublet-triplet splitting
• One pair of Higgsino doublets
• Hypercharge quantization
• Absence of strong CP violation

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The SM is written in 4D but superstring
iswritten in 10D. We need a mechanism to lower
the spacetime dimensions with a 6D internal
space and gauge group.

• Compactification with 6D Calabi-Yau space
• Orbifold compactification
  

Here we discuss the orbifold road. It is a simple
strategy in string compactification. Recently it
is also applied in field theoretic
compactification.
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Let us look for string compactification which
enable the spectrum go through spontaneous
symmetry breaking. Namely, allow the
possibilities having the needed Higgs particles.

• E8?E8 or SO(32) (10D theory)
• 4D GUTs or SM (orbifold compactification at
  string scale)
• SU(5), SO(10), E6, SU(3)3 (GUT scale)
• SU(3) ? SU(2) ? U(1)(SUSY broken at MSUSY)
• SU(3) ? SU(2) ? U(1)Y (string theory, gravity)
• SU(3) ? U(1)em (at MZ)

Below the orbifold compactification scale, we
need the presence of Higgs fields allowing the
needed symmetry breaking.
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2. Orbifold geometry
A naïve toroidal compactification does not lead
to chiral fermions. Orbifold compactification
circumvent this unsatisfactory feature via a
boundary condition on the manifold. An orbifold
is obtained by identifying points under a
discrete symmetry group S. This space group has
elements (?,v)
The torus is the most simple orbifold with
elements (1,v). The orbifold is the torus modded
out by ?. The point group is S up to lattice
translation, S/? .
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shift v
?
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Fixed points
The orbifold is the modded out torus by ?. This
set of points is called the fundamental region.
In this fundamental region, there are fixed
points.
After a point group action ?, a fixed point comes
back to the original location
The number of fixed points is given by Lefschetz
fixed point theorem
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Z2 orbifold in 1D
Fundamental region (0, ?R)
Z2x Z2 orbifold in 1D
In FTO, one Z2 is used to break 5d N1 down to 4d
N1. Another Z2 is used to break SU(5) to the SM.
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Then, we have wave functions
Only the (,) modes can contain massless states.
Gauge and 2(55)
In addtion, mass- less fields can be put at fixed
points. Anomaly should be zero.
String orbifold is like having this kind of zero
masses.
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Two dimensional orbifolds
Topology of 6D internal space is studied most
non-trivially in three two tori. The easiest 2D
orbifold example is T2/Z2,
The number of fixed points is 4. By the
Lefschetz fixed point theorem, the fixed point
number is
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T2/Z3 Orbifold
The most widely studied orbifold is T2/Z3,
The matrix form of Z3 action is ?. By the
Lefschetz fixed point theorem, the fixed point
number is 3.
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T2/Z6 Orbifold
The T2/Z6 orbifold is
The fixed point number depends on the twists as
shown in Z6. This is the case in non-prime
orbifolds such as Z4, Z6, Z8, Z12.
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3. Application in string theory
As we constructed the 10D heterotic string, we
compactify 6 right movers and 616 left movers,
xi
Strings are extended objects. For a closed
string, we need the BC,
X(??) X(?)2?RL
In addtition, as the wave function in QFT, the
string can be twisted.
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Twisted string
Closed strings can arise with three varieties
Because of the orbifold geometry, we have a
holonomy for a complexified bosonic string,
going along the string
The allowed mode expansion is(similarly for
anti-holomorpic)
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The commutation relations of oscillators are
The oscillator number is
which can be fractional. ?a-1/30? can have the
oscillator number 1/3, because of the fractional
number on RHS of the commutators. The massless
string states must satisfy M2LM2R M2/2 with
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The left and right contributions to the zero
point energy, for a real bosonic degree, is
Thus, the bosonic left-movers have
Similarly, the right movers have
UT?0
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Z3 orbifold
The internal space shift is
27 fixed points. But the partition function
technique is more straightforward in nonprime
orbifolds.
The group space is also moded out by Z3. The
standard embedding which has the same shift in
the group space is
UT
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The massless weights P of UT are
78 of E6 together with H. Gauge group is SU(3)E6.
(3,27) of E6
So far we considered ML20. For the right movers,
we can consider R or NS. For the R sector, the
spinorial state transformation is
P(3,27), s3 Thus, matter Mult.3 from
UT Left-handed
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Twisted sector
(1,27) of of E6
There are more states with a novanishing
oscillator
Study of right movers allow only (----)
Left-handed Mult.fixed point number times
oscillator number.
(3,1) of of E6
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Anomaly cacellation
The representations we obtain are left-handed.
Checking SU(3) anomaly, we have
Anomaly free!!
3 (3,27) 27 (1,27) 81 (3,1)
Wison lines
So far we discussed the original DHVW. But we can
have (internal) gauge fields wrapping the torus.
Under the gauge transformation, a wave function
can acquire a phase in the gauge group space
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The gauge invariant measure of this is Wilson
lines
This gauge transformation is interpreted as an
embedding of Z3 in the group space. This in
effect transforms the weight P as
Consideration of this additional shifts makes it
possible to reduce the gauge group further.
Conditions for the Wilson lines are similar to
but not the same as V.
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All these considerations will be discussed in
a Z12 example in the end.
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4. Phenomenology
There are notable problems in the orbifold
compactification

• Chirality
• N1 SUSY
• Standard model SU(3)xSU(2)xU(1)
• Adjoint problem
• (v) Three families
• (vi) Hypercharge quantization
• (vii) Doublet-triplet splitting
• (viii) Flavor problem

As we have seen, the chiality and N1 SUSY
problems are easily resolved in orbifold
compactification. It is too much chiral.
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Standard-like models
(iv),(v) simple Z3 orbifold which has multiples
of 3
The shift V and Wilson lines directly producing
SU(3)xSU(2) are needed. This is achieved by
IKNQ showed the possibility of standard-like
models and doublet-triplet splitting. There are
many works along this line, including non-prime
orbifolds and fermionic constructions.
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U(1) charges
This kind of standard-like models has the sin2?W
problem which is in most cases smaller than 3/8.
For U(1) generators Qi, the untwisted secor
weight P? has eigenvalues
For twisted sectors the eigenvalues are
U(1) charges can be determined by the UT spectrum.
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In GUTs, the weak mixing angle is given by
For Q1 and Q2,
This is possible only when Qem gets contributions
from the first 5 entries only, which is not the
case in most standard-like models. If there are
other Qs contributing, C2 5/3.
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One typical published example is (Kim, 1988)
This gives the following hypercharges for a
standard model direction
Another example(FIQS) has Y(1/3)Q1-(1/2)Q2 Q4 .
It gives C211/3 and sin2?W3/14.
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Anomalous U(1)
For some orbifolds, there results an anomalous
U(1). One anomalous U(1)X from 10D heterotic
string is allowed since the Green-Schwarz
mechanism removes this U(1)X at string scale. At
low energy, there does not appear any anomalous
gauge symmetry.
In some cases, it is actually helpful, since the
D-term for this anomalous U(1) at string scale is
Then, for some negative values of Qi(X), the
corresponding scalars can develop VEVs, breaking
some U(1) symmetries.
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Global symmetries
For global symmetries, we try to introduce n and
n global charges at the end points of open
strings. This U(n) global symmetry is a symmetry
in the 2D world-sheet. But in the 10D target
space, this symmetry is corrdinate(Xi)
dependent.? local symmetry. Global symmetry is
difficult if not approximate, dictated by
discrete symmetries.
Model-independent axion gauge field Bµ? This
obtain mass by GS
Then, there appears a global symmetry below the
string scale. This can be used as a PQ symmetry
at low energy.(Kim) But if we introduce a
confining hidden sector gauge group, we must
consider two axions. Here, difficulty or bonus!!
Model-dependent axions can help necessarily a
quintessencial axion.
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Electroweak CP
One may consider that the electroweak CP is a
discrete symmetry in string theory or appears as
a subgroup of continuous symmetries. Whatever it
is, it arises from the compactification process
from 10D to 4D. If complex Yukawa couplings are
introduced then one can obtain a KM type CP
violation.
Even though the resulting Yukawas are real, the
effective theory seen at the electroweak scale
must give the KM type complex Yukawa couplings.
This is achieved by a spontaneous symmetry
breaking at high energy a la Nelson-Barr. But
then wormholes are interfering.
CKN studied the wormhole effect with the
embedding of CP in gauge group discrete gauge
smmetry. It is possible if the discrete gauge
symmetry is broken at small parameter.
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µ and doublet-triplet splitting
The orbifold action is embedded in the group
space. In fact this was the reason for obtaining
a smaller gauge group. If the embedding
distinguishes the color SU(3) and weak SU(2),
there is a possibility of distinguishing colored
partners from weak partners. Thus, it is possible
to distinguish color triplets and weak doublets
of the Higgs quintet doublet triplet splitting.
(IKNQ, Kawamura)
It is also related to the µ problem since it
starts with vanishing µ, and generates it at the
electroweak scale,
In Z3 orbifold, UU and TT are not allowed. Thus,
the mu term is not appearing from
compactification. But it is not the whole story.
SH1H2 can appear and if S gets a large VEV, the
mu problem is not understood. Somehow, a symmetry
such as PQ or R may be needed to forbid a large
mu. Then, at the intermediate scale a correct
magnitude is generated,(CKN)
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sin2?W and GUT
The standard-like models suffer from the sin2?W
problem. In this regard, GUTs are very
attractive. However, simple GUTs have the adjoint
problem. The Kac-Moody algebra with level k1
does not allow adjoint representation. Obtaining
the SM from E6, SO(10), and SU(5) needs an
adjoint representation. These GUTs are not
allowed from string if level 1 spectrum is the
only possibility. Higher level algebra may
introduce exotic(color 6 for example) Quarks.
Finding out models without exotic quarks with
higher level algebra depends on luck.
HESSNA (hypercharge embedding in semi-simple
groups, not needing adjoint representation)
trnification SU(3)3, Pati-Salam SU(4)xSU(2)xSU(2)
Trinification is possible only from Z3 orbifold.
All Z3 orbifolds with two Wilson lines are
tabulated. Trinification is similar to E6, but
allows the possibility of spontaneous symmetry
breaking. Pati-Salam is very similar to SO(10)
and allows SSB.
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A schematic view of the evolution of gauge
couplings is expected as below. Actually, one
needs information of all 4D fields below the
string scale.
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5. Construction example with Z12
Consider Z12-I.
We use the Kac-Peterson-Choi-Kim method of using
Dynkin Dia.
Applying a few Weyl transformations, we have
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The gauge goup is SU(5)xSU(3). Summary of UT
fields are
For twisted sectors, the masslessness condition ,
for the left and right movers, is
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The twisted sector massless particles are
For the 4th twisted sector, let us show some.
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(No Transcript)
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The number of zero modes is calculated blindly
by the following partition function, i.e. by
picking up the appropriate combinations of wave
functions.
The ? is chosen from the table
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Now the problem is to calculate the phase ?. It
is given by
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Thus the total spectrum is
Anomaly check SU(3) (21)U (-2-18-65)T
0 SU(5) (-3-33)U (-3326-5)T 0
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Summary
We have listed several theoretical problems which
can be answered in models beyond the SM. The most
conspicuous one is the chirality problem.
Toroidal compactification does not give
the chirality but orbifolding it allows the
chirality. Then, we have reviewed possibilities
of resolving the outstanding theoretical
problems in orbifold compactifications. The most
widely studied one is Z3, but recently, nonprime
and product obifolds are also studied. Why not
the simplest orbifold? Now we are trying to find
at least one correct vacuum.
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We will discuss this route, from h-string
10-30 cm
Our universe with many particles 1088
pp
SM l ? W? Z q gluons
plus vacuum energy
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