String scale unification in an SU(6)xSU(2) GUT

17 Pages·0.2 MB·English

by  J. Rizos

SISSA/173/96/EP
hep-ph/9702295
STRING SCALE UNIFICATION IN AN SU(6)×SU(2) GUT
8 J. RIZOS1,2 and K. TAMVAKIS2
9
9
1
1International School for Advanced Studies, SISSA
n
a Via Beirut 2-4, 34013 Trieste, Italy
J

6
2
2Physics Department, University of Ioannina
3
Ioannina, GR45110, GREECE
v
5
9
2
2
0 Abstract
7
9
Weconstruct andanalyzeanSU(6)×SU(2) GUT.The modelisk=1 stringembedable inthe
/
h
sense that we employ only chiral representations allowed at the k=1 level of the associated
p
- Kaˇc-Moody Algebra. Both cases SU(6)× SU(2)L and SU(6)× SU(2)R are realized. The
p
model is characterized by the SU(6)×SU(2) → SU(4)×SU(2)×SU(2) breaking scale M ,
e X
h and the SU(4) × SU(2) × SU(2) → SU(3) × SU(2) × U(1) breaking scale M . The
C L Y R
:
v spectrum bellow M includes an extra pair of charge-1/3 colour-triplets of mass M ≤ M
R I R
i
X that does not couple to matter fields and, possibly, an extra pair of isodoublets. Above M
X
r the SU(6) and SU(2) gauge couplings always unify at a scale which can be taken to be
a
the string unification scale M ∼ 5×1017GeV. The model has Yukawa coupling unification
s
since quarks and leptons obtain their masses from a single Yukawa coupling. Neutrinos
obtain acceptably small masses through a see-saw mechanism. Coloured triplets that couple
to matter fields are naturally split from the coexisting isodoublets without the need of any
numerical fine tuning.
SISSA/173/96/EP
February 1997
1.Introduction.
Superstring Theory [1] is at present our best candidate for a unified theory of all particle
interactions being a consistent theoretical framework that incorporates quantum gravity
and supersymmetric gauge theories [2]. Unification takes place at energies close to the
Planck scale while at lower energies gauge interactions are described in terms of an effective
field theory whose spectrum consists of the massless string modes. From the low energy
point of view the Standard Model and its N = 1 supersymmetric extension, the Minimal
Supersymmetric Standard Model (MSSM), can be naturally embedded in a Grand Unified
Theory (GUT) with interesting phenomenological and cosmological consequences. GUTs
[3] can successfully predict sin2θ , fermion mass relations, charge quantization as well as
W
provide a mechanism to explain the baryon asymmetry of the Universe [4]. However, in the
context of quantum field theory no severe restrictions exist on the gauge group or the matter
content of a GUT apart from the requirement that it should accommodate the MSSM. Thus
a lot of possibilities arise including minimal SU(5) and its extensions [5], varieties of SO(10)
[6], E(6) models [7] e.t.c.
Accommodating a GUT in the framework of Superstring Theory 1, or more precisely
assuming that the GUTisthe lowenergy effective field theoryof a fourdimensional heterotic
superstring compactification, imposes serious restrictions on the GUT spectrum. For gauge
groups realized at level k = 1 of the World-Sheet Affine Algebra, only the chiral multiplets in
the vector and antisymmetric representations of SU(n) groups and the vector and spinor of
SO(n) groups are massless. The absence of adjoint scalars limits the possibilities of breaking
to the MSSM through the Higgs mechanism and diminishes the number of candidate GUT
models. Apart from this serious restriction, superstrings offer a new possibility. The GUT
gauge group does not have to be simple in order to guarantee unification. Semi-simple or
product groupsareequally acceptablesince stringtheorytakes over thejobofgaugecoupling
unification.
HistoricallythefirstGUTsthatwerestudiedascandidatelowenergysuperstring effective
models were the maximal E(6) subgroups [8] as they arise naturally in the context of Calabi-
Yau compactifications [9, 10, 11]. A typical example of such a model is the SU(3)×SU(3)×
SU(3)GUT[12]whileotherpossibilitiesasSU(6)×U(1)havealsobeenconsidered[13]. Soon
it was realized that SO(10) subgroups were also possible in the context of Orbifolds [14, 11]
or in the free-fermionic formulation [15]. Two typical GUT examples, namely the “flipped-
SU(5)” [16] and the supersymmetric Pati-Salam model [17], were explicitly constructed in
the framework of the Fermionic Superstrings [18, 19]. Standard Model-like alternatives have
also been explored [20].
Recently, a new the possibility has been explored , namely GUTs realized at the k ≥ 2
level of heterotic superstring compactifications [21]. Some generic features of these models
have been presented in [22, 23]. Higher level models allow the presence of adjoint chiral
superfields and thus the number of the candidate GUT gauge groups is enhanced. Although
some explicit superstring models have been presented [24], there are still some problems to
1ThetermGUTisusedhereinaloosesenseforagaugemodelwithgaugegroupG⊃SU(3)×SU(2)×U(1)
which accomodates the MSSM particles and displays partial unification and/or correlation of the standard
model group gauge factors.
1
overcome [22, 23, 25]. Such is the absence of certain couplings. For instance, the remnants of
the GUT breaking adjoint Higgs, an isotriplet and a colour-octet, remain light to all orders
in perturbation theory [22, 25].
Another issue raised in the framework of the Superstring embedding of the MSSM is that
of the “unification mismatch”. Assuming MSSM to hold, low energy data indicate gauge
coupling unification at a scale M ∼ 1016GeV [26] more than one order of magnitude below
X
the string scale M ∼ 5×1017GeV which is thetypical string unificationscale. The prospects
s
to fill this gap by threshold corrections due to the infinite tower of massive string modes
[27, 28] seem rather restricted [29] as explicit calculations in superstring models show that
such thresholds are very small [30]. This could be taken as an indication for the existence of
intermediate scalesinwhichextramatterstatesbecomemassiveandtheirthresholdsincrease
the coupling unification scale up to M [31, 32, 25]. One is thus motivated to incorporate
s
this additional feature in a candidate GUT model.
In the current state of elaboration of String Theory, the string vacuum is not uniquely
determined and the various gauge groups appear on equal footing. It is hence interesting,
from the low energy point of view, to classify all gauge models satisfying the known string
constraints and see whether they can meet the criteria imposed by low energy data and at
the same time look for signatures that distinguish among them.
As is well known, all quark and lepton fields can be accommodated in the 27 represen-
tation of E(6) together with a pair of isodoublets suitable to serve as electroweak Higgses
and a pair of charge ±1 colour-triplets. Among the maximal subgroups of E(6) we find
3
SU(6)×SU(2), a regular subgroup which apart from [33] has not received much attention
in the literature. Decomposing the 27 under SU(6)×SU(2) we get [34]
27 = (6,2)+(15,1)
where 15 is the two-index antisymmetric representation of SU(6). In what follows we shall
construct a k = 1 string embedable SU(6) × SU(2) GUT using chiral superfields in the
(6,2) and (15,1) representations only. Two distinct ways of embedding the standard model
group in SU(6) × SU(2) arise depending on whether we identify the electroweak SU(2)
L
with the GUT SU(2) factor or with the SU(4)× SU(2) subgroup of SU(6). In the latter
case we have an SU(6)×SU(2) model while in the former an SU(6)×SU(2) model. Both
R L
cases SU(6)×SU(2) and SU(6)×SU(2) are realizable. Apart from the GUT scale M
L R X
of SU(6) breaking, the model is also characterized by the extra scale of SU(2) breaking
R
M . Below the scale M the spectrum contains an extra pair of coloured triplets of mass
R R
M ≤ M . The SU(6) and SU(2) gauge couplings always meet at energies above M
I R X
introducing an additional scale which can always be taken to be the string unification scale
M ∼ 5×1017GeV. In addition these models have some other very interesting features. The
s
electroweak Higgses, together with extra charge-1/3 colored triplets, participate in the same
representations that contain quarks and leptons. These isodoublets and colour-triplets also
come in family replicas. Despite that, the triplet-doublet splitting can be achieved naturally
without the need of any numerical fine tuning. Quark and lepton masses arise from a
single Yukawa coupling, a cubic interaction of matter superfields, just as in E , despite the
6
fact that matter fields come in a reducible representation. Thus, SU(6) × SU(2) displays
2
Yukawa coupling unification. In the case of one pair of Higgs isodoublets the mass relations
m cotβ = m = m are true and imply strong constraints on m , tanβ [36] and the various
t b τ t
supersymmetry breaking parameters [37].
2. Symmetry breaking and model building.
Decomposing the 15 representation under SU(4)×SU(2)×U(1) ⊆ SU(6) we obtain
15 = (1,1,4)+(6,1,−2)+(4,2,1)
Thus, it is clear that the GUT symmetry breaking SU(6)×SU(2) → SU(4)×SU(2)×SU(2)
can be achieved with a non-zero v.e.v. of (15,1)+(15,1) in the D-flat direction h(1,1,4)i =
h(1,1,−4)i . The decomposition of the adjoint 35 of SU(6) under SU(4)×SU(2)×U(1)
takes the form 35 = (1,1,0) + (1,3,0) + (15,1,0) + (4,2,−3) + (4,2,3). This helps to
see that the tetraplets in 15 will be absorbed while the sextets (6,1,−2) , (6,1,2) and the
singlet combination (1,1,4) + (1,1,−4) will survive the Higgs phenomena. Nevertheless,
cubic superpotential couplings
(15)3 = (6,1,−2)2(1,1,4)+(4,2,1)2(6,1,−2)
while not spoiling F-flatness, can give the sextets a mass. The first stage of symmetry
breaking being as described, two possibilities arise as we move further down.
a) An SU(6)×SU(2) model.
L
The desired breaking pattern down to the standard model gauge group is
M M
X R
SU(6)×SU(2) SU(4)×SU(2) ×SU(2) SU(3) ×U(1) ×SU(2)
L R L C Y L
−→ −→
As we described above, the first stage of symmetry breaking requires a pair of Higgses 2
H(15,1)+H(15,1) = (N ) +(∆c +∆ )
H (1,1,1) H H (6,1,1)
+(Ec +Nc +Dc +Uc ) +conj.reps. (1)
H H H H (4,2,1)
AD-flatv.e.v. hN i = hN i = V breaksSU(6)×SU(2) toSU(4)×SU(2) ×SU(2) . The
H H X L R L
surviving statesarethesextets(∆c +∆ ) ,(∆c +∆ ) plusasingletcombination
H H (6,1,1) H H (6,1,1)
N +N .
H H
The second stage of symmetry breaking down to the standard model gauge group can be
achieved introducing an additional pair
H(15,1)+H(15,1) = (n ) +(δc +δ )
H (1,1,1) H H (6,1,1)
+(ec +νc +dc +uc ) +conj.reps. (2)
H H H H (4,2,1)
2We introduce here a mixed notation where the SU(4)×SU(2)R×SU(2)L ⊂SU(6)×SU(2) quantum
numbers are shown explicitly while the symbols for the fields indicate Standard Model quantum numbers
in the usual notation : ∆c,Dc,δc → (¯3,1,1), Uc,uc → (¯3,1,−2), Ec,ec → (1,1,1), N,n,νc → (1,1,0),
3 3
h,L,η →(1,2,−1) etc.
2
3
A D-flat v.e.v. hνc i = hνc i = V breaks SU(4) × SU(2) down to SU(3) × U(1) .
H H R R C Y
The nine surviving states are the sextets (δc +δ ) , (δc +δ ) and the singlets
H H (6,1,1) H H (6,1,1)
(n ) , (n ) , which do not participate in the symmetry breaking anyway, a pair of
H (1,1,1) H (1,1,1)
colour-triplets dc , dc and a combination of singlets νc +νc .
H H H H
The Higgs sector superpotential, suitable for the desired symmetry breaking pattern will
be taken to be the cubic superpotential (invariant under the discrete symmetry H → −H)
W = λHH2 +λHH2 (3)
H
which leads to the Higgs mass terms
(W ) = λV δc δ +λV dc ∆ +λV δc δ +λV dc ∆ +... (4)
H mass X H H R H H X H H R H H
Two pairs of coloured triplets, namely δ , δc and δ , δc , obtain a mass of O(M ) and two
H H H H X
pairs, namely dc , ∆ and dc , ∆ , obtain a mass of O(M ). Below M , apart from the four
H H H H R R
singlet states n , n , N +N and νc +νc , the renormalizable superpotential interactions
H H H H H H
in (3) leave the pair of coloured triplets ∆c , ∆c massless. Effective non-renormalizable
H H
terms could be added to (3) which would generate masses for these states. These could arise
due to the exchange of massive states present in all string constructions. Their exact form
depends on the details of the specific string model. A necessary constraint on these terms
is the requirement that they should not spoil F-flatness. Note however that violations of
F-flatness that lead to scalar masses of O(TeV) can be tolerated since the supersymmetry
breaking is of that order. This implies hFi/M ≤ O(TeV) or hFi ≤ (1011GeV)2. High
order non-renormalizable terms of the form (HH)n/M2n−3 lead to F ∼ (M /M)2n−1M2
X
and satisfy this constraint if (M /M)2n−1 ≤ 10−14. For M /M ∼ 0.01 this corresponds to
X X
n ≥ 4. Writing down the contents of such a term, gives
(V )2n−2
∆W = (HH)n/M2n−3 = X (∆ ∆ +∆c ∆c +...) (5)
H M2n−3 H H H H
This corresponds to an intermediate mass M ∼ (M /M)2n−2M for the triplets ∆c , ∆c
I X H H
and the singlet N + N . The generated intermediate mass is constrained to be M ∼
H H I
(M /M)2n−2M ≤ O(10−14)M2/M . For M /M ∼ 0.01, this implies M ≤ O(106GeV).
X X X I
For M /M ∼ 0.1, M ≤ O(105GeV). Note also that, although the product in (5) can arise
X I
either as (15) (15)ij or as (15) (15) (15) ǫijklmn(15)pq(15)rs(15)tzǫ , only the first
ij ij kl mn pqrstz
case gives rise to masses. It should be remarked however that the exact form of the non-
renormalizable terms depends onthe detailsof the, possibly, underlying string modelandthe
above presented mechanism that supplies the remaining colour-triplets with an intermediate
mass, although perfectly consistent in the framework of the GUT, serves as an existence
proof of phenomenologically desirable scenaria that can be realized within the string model.
Summarizing, weseethatintroducingtheHiggsesH, H,H , H wecanachieve thesuccessive
breaking SU(6) × SU(2) → SU(4) × SU(2) × SU(2) → SU(3) × U(1) × SU(2) .
L R L C Y L
Below M the two pairs of Higgses contribute to the spectrum of the model with two sextets
X
(∆ +∆c ) and conjugate states, two tetraplets (uc +dc +ec +νc ) and conjugate
H H (6,1,1) H H H H (4,2,1)
states, and the three singlets n , n , N +N . Below M only the coloured triplets ∆c ,
H H H H R H
4
∆c survive, together with four singlets n , n , N +N , νc +νc . The coloured triplets
H H H H H H H
obtain a mass at the lower scale M .
I
Quarks and leptons can be accommodated in the (15,1)+(6,2) representation in three
family replicas
φ(15,1) = N +(δc +δ) +(ec +νc +dc +uc) (6)
(1,1,1) (6,1,1) (4,2,1)
ψ(6,2) = (h+hc) +(q +l) (7)
(1,2,2) (4,1,2)
Notice that together with quarks and leptons we have pairs of isodoublets, suitable as elec-
troweak Higgses h , hc and coloured triplets δ , δc in three family replicas too. We next
introduce the matter self coupling
W = Y φ ψ ψ
M ijk i j k
1
= Y˜ (q dch +q uchc +l ech +l νchc + δ q q +δcq l +N h hc) (8)
ijk j i k j i k j i k j i k 2 i j k i j k i j k
where Y˜ = Y +Y . Quark and lepton masses arise exclusively from the single Yukawa
ijk ijk ikj
coupling Y˜ . The fifth and sixth term in (8), together with a δδc mass term for the extra
ijk
colour-triplets, give rise to the D = 5 operators qqql that violate baryon number and can
induce proton decay [35]. These operators are controllable if the δ δc mass is of O(M ).
i j X
This can be achieved if we introduce the couplings
1 1
∆W = λ Hφ φ + λ′ Hψ ψ (9)
M 2 ij i j 2 ij i j
which give the terms
λ (N δ δc +ecuc∆ +νcdc∆ +ucdc∆c +...)
ij H i j j j H i j H i j H
+λ′ (N h hc +q q ∆ +q l ∆c ) (10)
ij H i j i j H i j H
Masses of O(λV ) and O(λ′V ) correspondingly can now arise for the coloured triplets and
X X
isodoublets. Note that both λ and λ′ couplings are symmetric and λ′ must have at
ij ij ij
least one zero eigenvalue. Baryon number violating operators of D = 5 of the type ecucucdc,
νcucdcdc andqqql would also appearif a mass term∆ -∆c were present. This is notthe case
H H
however since ∆c mixes only with ∆c which does not mix with matter fields. Nevertheless,
H H
D = 6 operators of the form λ λ′ ucdcq†l† , generated by the exchange of a scalar triplet
ij km(cid:16) i j k m(cid:17)
∆ or∆c , are stillpossible andcanbedangerous inthe case ofanintermediate triplet mass.
H H
The presence of these operators depends on the structure of the coupling matrices λ λ′ .
ij km
For instance, in the case of diagonal couplings, a simple condition like λ = 0,j = 1,2 would
1j
be sufficient to render these operators harmless without affecting the massiveness of δ ,δc.
i i
In a general situation this problem can be evaded by requiring a more complicated texture
structure for λ and/or λ′ . Another more drastic solution is to set λ′ = 0. In this case
ij km km
no such operators exist but all three pairs of isodoublets are left massless. This is not a big
problem since their masses can be generated by non-renormalizable terms. For example the
term λ′i′jH4ψ ψ leads to a λ′′ <νHc >4h hc isodoublet mass matrix without introducing any
M3 i j ij M3 i j
unwanted couplings.
5
Expressing the Yukawa interactions in terms of the Higgs mass eigenstates, we obtain
Ya(q dch˜ +l ech˜ +q uch˜c +l νch˜c ),a = 1,2,3 (11)
ij i j a i j a i j a i j a
It is clear that in the case of only one pair of massless Higgses, since there is a single,
generation dependent, Yukawa coupling, we have the fermion mass relations m = m cotβ =
b t
m which imply the prediction of m [36] as well as strong constraints on tanβ (tanβ ≥ 40)
τ t
and the susy-breaking parameters [37]. In the case that a second pair of isodoublets is
massless and obtain v.e.v.’s, there is no simple mass relation unless there is v.e.v. alignment.
Depending on the details of the specific string model or possibly existing family symmetries,
the Yukawa couplings for the light generations could arise as effective non-renormalizable
terms with a suppressing factor (hSi/M)n involving the v.e.v. of a singlet field S [41].
Nevertheless, an alternative to such a scenario would be to keep more doublets light and
imposeahierarchyontheavailablemanyv.e.v’s. Non-renormalizabletermscanalsogenerate
the required large Majorana mass for the right-handed neutrino. Introducing
Y′′φ φ H2/M (12)
ij i j
we are only led to
Y′′V2
Y′′νcνchνc i2/M = ij Rνcνc (13)
ij i j H M i j
Depending on the effective non-renormalizable coupling Y′′ this term can lead to acceptable
ij
neutrino masses through a see-saw mechanism.
b) An SU(6)×SU(2) model.
R
The first breaking SU(6) × SU(2) → SU(4) × SU(2) × SU(2) occurs exactly as
R L R
in the previous model using a pair of Higgses (15,1) + (15,1) . Nevertheless, due to the
different positionofSU(2) their content looksdifferent interms ofstandardmodelquantum
L
numbers. Again we introduce
H(15,1)+H(15,1) = (N ) +(∆ +∆c ) +(L +Q ) +conj.reps. (14)
H (1,1,1) H H (6,1,1) H H (4,2,1)
A D-flat v.e.v. hN i = hN i = V breaks the gauge group to SU(4)×SU(2) ×SU(2) .
H H X L R
The coloured triplets ∆ , ∆c , ∆c , ∆ and a combination of singlets, namely N +N ,
H H H H H H
survive the Higgs phenomena.
The second stage of symmetry breaking down to the Standard Model group is achieved
introducing
H(6,2)+H(6,2) = (η +ηc ) +(dc +uc +ec +νc ) +conj.reps. (15)
H H (1,2,2) H H H H (4,1,2)
AD-flatv.e.v. hνc i = hνc i = V breaksthegaugegroupdowntoSU(3) ×SU(2) ×U(1) .
H H R C L Y
In addition to η , ηc , η , ηc which are not affected by the Higgs phenomena, dc , dc
H H H H H H
and the singlet νc +νc survive.
H H
The Higgs sector superpotential will be taken to be again a cubic one of the form
W = λHH2 +λHH2 (16)
H
6
which leads to the Higgs mass terms
(W ) = λV η ηc +λV ∆ dc +conj.reps. (17)
H mass X H H R H H
Two pairs of Higgs isodoublets become massive at M and two pairs of coloured triplets,
X
namely ∆ , dc and ∆ , dc become massive at M . Below M we are left with a massless
H H H H R R
pair of colour-triplets ∆c , ∆c . Following the philosophy of the previous model we can
H H
expect that non-renormalizable terms (HH)n/M2n−3 will give rise to a mass of intermediate
value M ∼ 106GeV or smaller for the pair ∆c , ∆c as well as for the singlet N +N .
I H H H H
The matter fields are introduced as
φ(15,1) = N +(δ +δc) +(l+q) (18)
(1,1,1) (6,1,1) (4,2,1)
ψ(6,2) = (h+hc) +(dc +uc +ec +νc) (19)
(1,2,2) (4,1,2)
in three family replicas. The Yukawa terms are obtained from
W = Y φ ψ ψ (20)
M ijk i j k
= Y˜ (l ech +l νchc +q dch +q uchc +N h hc +δ dcνc +δ ucec +δcdcuc)(21)
ijk i j k i j k i j k i j k i j k i j k i j k i j k
Let us now introduce the Higgs-matter interactions
∆W = λ Hφ φ +λ′ Hψ ψ (22)
M ij i j ij i j
= λ (N δ δc +∆ q q )+λ′ (N h hc +∆ dcνc +∆ ucec +∆c dcuc)
ij H i j H i j ij H i j H i j H i j H i j
Note that the couplings λ , λ′ are symmetric. These couplings are sufficient to render
ij ij
all triplets δ , δc massive with a mass of O(M ). Of course λ′ has to be restricted in
i i X ij
family space in order to obtain the desired number of massless pairs of electroweak Higgses.
The D = 5 operators that violate baryon number and can arise from W are going to be
M
sufficiently suppressed if δ , δc have a mass of O(M ). On the other hand, since ∆ and
i i X H
∆c do not mix, there is no danger of any D = 5 operators arising from ∆W . Note that
H M
∆ obtains its mass from dc which does not mix with matter. D=6 operators involving the
H H
exchange of a ∆ scalar are also possible and could be dangerous for low M . The related
H R
discussion in the previous model applies also here. If the condition λ′ = 0 is imposed in
ij
λ′′
order to circumvent this problem, the non-renormalizable term ijH4ψ ψ can be invoked
M3 i j
<νc >4
to generate H masses for the unwanted doublets.
M3
Finallythe Yukawa interactions areidentical tothoseof theprevious modelandallpoints
concerning Yukawa coupling unification and mass relations are the same. Again, in order to
obtain acceptable neutrino masses, the non-renormalizable interactions
Y′′ψ ψ HH/M
ij i j
have to be invoked. This term generates a Majorana mass matrix for the right-handed
neutrinos of order Y′′V2/M .
ij R
3. Renormalization group analysis.
7
As follows from the previous analysis the SU(2) scale M is a free parameter while
R R
the triplet mass scale M depends on the details of the model and in particular of the non-
I
renormalizable contributions in the superpotential. In addition the SU(6)×SU(2) model,
L
denotedasmodel(I)fromnowon, andtheSU(4)×SU(2) model, denotedasmodel(II),are
R
indistinguishablebellowM wherewehaveanSU(3) ×SU(2) ×U(1) theorywithanextra
R C L Y
charge-1/3 colour-triplet pair ofintermediate mass M . In what follows we shallassume both
I
M and M as free parameters and derive the constraints imposed on them when demanding
R I
SU(6) and SU(2) coupling unification at the string scale M = 5.27×g ×1017GeV. For
S S
simplicity we consider here only the case of one or two (N = 1,2) massless isodoublet pairs
H
below M and leave the more complicated analysis of extra intermediate mass doublets for
R
a future publication [38]. Integrating the renormalization group equations from M up to
Z
M , we obtain
R
˜
b M b M
α−1(M ) = α−1 + i ln( Z)+ i ln( I ) (23)
i R i 2π M 2π M
R R
Where i = 1,2,3 and α , α , α stand for the values of the three gauge couplings at M .
1 2 3 Z
The RG coefficients take up the values b = −3 , b = N , b = 6+ 3NH , ˜b = 1 , ˜b = 0
3 2 H 1 5 3 2
,˜b = 2 At an energy scale µ ≥ M we have an SU(4)×SU(2) ×SU(2) theory in both
1 5 R R L
cases (I) and (II). Note that the particle content for (I) and (II) is the same. The three
gauge couplings are given by
b M
α−1(µ) = α−1(M )+ i ln( R) (24)
i i R 2π µ
Where i = 4, 2R, 2L. Thecoefficients takeupthevaluesb = −2 , b = N , b = 4+N .
4 2L H 2R H
The matching conditions at M are
R
α (M ) = α (M ) (25)
4 R 3 R
5 2
α−1(M ) = α−1(M )− α−1(M ) (26)
2R R 3 1 R 3 3 R
α (M ) = α (M ) (27)
2L R 2 R
As µ moves away from M two possibilities arise. If α meets first with α , we have model
R 2R 4
(I). In the opposite case of α meeting first with α , we have model (II). From the point of
2L 4
view oftherenormalizationgroupanalysis, thelowenergy dataaswell asthefreeparameters
of the theory, M and M , determine which case is realized.
R I
Let us first study the case of model (I) which is defined by
α (M ) = α (M ) = α (M ) ≥ α (M ) (28)
6 X 2R X 4 X 2L X
M stands for the SU(6) × SU(2) breaking scale. Manipulating all previously displayed
X
equations, we obtain
1 M 5 4 M 1 M
ln( X) = (α−1 −α−1)− ln( R)− ln( I ) (29)
2π M 3(6+N ) 1 3 π(6+N ) M 2π(6+N ) M
Z H H Z H Z
The inequality in (28) defining model (I) is equivalent to
5 2 2 M 4 M
F = (2+N )α−1+ (4−N )α−1−(6+N )α−1+ ln( I )− (2+N )ln( R) ≤ 0 (30)
3 H 1 3 H 3 H 2 π M π H M
Z Z
8
(II)
15
N = 1
H
(I)
10
)
I
M (II)
(
0
1
g
o
l 5
N = 2
H
0
(I)
12 13 14 15 16
log (M )
10 R
Figure 1: The intermediate scale log (M ) in the case N = 1,2 as function of log (M ),
10 I H 10 R
for α = .11 (dashed line) and α = .13 (continuous line).
3 3
At a scale µ ≥ M the SU(6)×SU(2) gauge couplings are
X L
b M
α−1(µ) = α−1(M )+ i ln( X) (31)
i i X 2π µ
Where i = 6, 2L . The coefficients take up the values b = −1 , b = 3 . The fact that
6 2L
b < 0 and b > 0 together with (28) imply that there exists always a scale µ = M > M
6 2L S X
at which the SU(6) and SU(2) couplings become equal
α (M ) = α (M ) ≡ α (32)
6 S 2L S S
Solving for it, we obtain
1 M 1 1 M (2−N )
ln( S) = (α−1 −α−1)− ln( I )+ H ln(M /M ) (33)
2π M 4 2 3 8π M 8π X Z
Z Z
while the value of the unified gauge coupling is given by
1 M 1 M 1 M
α−1 = α−1 + ln S + ln X + ln I (34)
S 3 2π M 2π M 2π M
Z Z Z
The case of the model (II) is obtained for
α = α (M ) = α (M ) ≥ α (M ) (35)
6 2L X 4 X 2R X
9