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International JournalofModernPhysicsA
(cid:13)c WorldScientificPublishingCompany
5
0
0 A dynamical mechanism to explain the top-bottom quark mass
2 hierarchy
n
a
J
A.DoffandA.A.Natale
0
2 Instituto de F´ısica Te´orica, Universidade Estadual Paulista, Rua Pamplona 145, 01405-900,
S˜ao Paulo, SP, Brazil
4 doff@ift.unesp.br
v [email protected]
3
9
1 We discuss the mass splitting between the the top and bottom quarks in a technicolor
8 scenario. The model proposed here contains a left-right electroweak gauge group. An
0 extended technicolor group andmirrorfermions areintroduced. Thetop-bottom quark
4 masssplittingturnsouttobeintimatelyconnectedtothebreakingoftheleft-rightgauge
0 symmetry.Weakisospinviolationoccurswithintheexperimental limits.
/
h Keywords: Technicolor models; Unifiedtheories and modelsof strongand eletroweak ;
p Non-standard-model Higgsbosons.
-
p PACSnumbers:12.60.Nz,12.10.Dm,14.80.Cp
e
h
: 1. Introduction
v
i
X In the standardmodel of elementary particles the fermion and gauge bosonmasses
r aregenerateddue to the interactionofthese particles with elementaryHiggs scalar
a
bosons. Despite the success of the model there are some intriguing points as, for
instance,the enormousrangeofmassesbetweenthe lightestandheaviestfermionic
generations. In particular we can also quote the large ratio between the Yukawa
coupling constants Y /Y 1/41, required to give the correct masses to the top
b t
∼
and bottom quarks. These questions could be better explained with the introduc-
tion of new fields and symmetries. One of the possibilities in this direction is the
substitution of elementary Higgs bosons by composite ones in the scheme named
technicolor 1,2.
The beautiful characteristics of technicolor (TC) as well as its problems were
2
clearly listed recently by Lane . Most of the technicolor problems may be related
2
to the dynamics of the theory as described in Ref. . Although technicolor is a non-
Abelian gauge theory, it is not necessarily similar to QCD, and if we cannot even
say that QCD is fully understood up to now, it is perfectly reasonable to realize
the enormous work that is needed to abstract from the fermionic spectrum the
underlying technicolor dynamics.
The many attempts to builda realistic modelof dynamicallygeneratedfermion
masses are reviewed in Ref.1,2. Most of the work in this area try to find the TC
1
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2 A. Doff and A. A. Natale
dynamics dealing with the particle content of the theory in order to obtain a tech-
nifermionself-energythatdoesnotleadtophenomenologicalproblems,asanexam-
3
plewecanconsiderthe schemeknownaswalkingtechnicolor .Morerecentpapers
4
envisage such different dynamics as a consequence of a conformal symmetry . In
this work we propose a model to explain the large (t-b) splitting in the context of
a technicolor scenario.
In order to obtain R 1 , where R = mb, we will consider a technicolor model
∼ 40 mt
based on the gauge group
′
G=SU(3) G (1)
ETC ⊗
′
where in the above structure we define G =SU(3) SU(2) SU(2) U(1) .
′C ⊗ L ⊗ R⊗ S
In this model we consider that the gauge group G is embedded in a grand-unified
5
theory (GUT) with a gauge group SO(12) . This group contains the electroweak
6
Left-Right model that has been widely studied and naturally contains a complete
fermionicgeneration(Ψ)aswellasitsrespectivemirrorpartner(Ψ).Anexplanation
of why we have performed this particular choice will become clear in the fourth
section. Models along this line have been proposed by Appelqubist and Schrock7.
The variation in our case is the introduction of the mirror sector. An interesting
feature of our model is that we are able of the relate the top-bottom quark mass
splitting as ratio between Left-Right boson masses. The fermionic content of the
modelisdividedintotwodistinctsectors,thesector(a)containingtandbquarks,T
andBtechniquarksandtheirrespectiveleptons.Thesector(b),ormirrorsector( ),
contains the mirror partners tˆandˆb with respective mirror leptons.
In the above model we do not introduce an ETC gauge sector associatedto tbhe
mirrorfermions because the mass scale peculiar to these particles is expected to be
8
around(1-10)TeV .Thereforeinthisscenarioitcanbenaturaltoconsiderthatthe
mirror particles obtain their masses through the same mechanism responsible for
the SO(12) SU(3) SU(2) U(1) symmetry breaking. In this model only
→ C ⊗ L ⊗ Y
the usualfermions andLH gaugebosons willreceivetheir massesdynamically.The
mirrorfermionsandtheRHgaugebosonswillbeveryheavy,andtheirmassescould
result from the existence of elementary scalar fields which can naturally appear at
the GUT scale.
Toensurethatm =m withoutintroducinglargeviolationsofweakisospinwe
t b
6 9
will assume an scheme proposed by Sikivie et al. some years ago . The basic idea
ofthis mechanismis toputthe chiralcomponents(LHandRH)ofagivenfermion,
member of an electroweak doublet, in different ETC representations. For example,
in our model the chiral components of the bottom quark will be in different ETC
representations,therefore the bottom quark does not receive mass at leading order
as happens for the top quark.Howeverin our model diagramsasthe ones depicted
in the Fig.(1) will contribute at higher order to the bottom quark mass.
February 2, 2008 0:40 WSPC/INSTRUCTION FILE split
A dynamical mechanism toexplain the top-bottom quark mass hierarchy 3
2. The fermionic content of the model
9
According to the group structure G and following the recipe of Ref. we can now
choosethefermionicrepresentationinsuchawaythatthebottomquarkmassarise
fromdiagramsofthe type shownin Fig.(1).This pointwill be discussedwith more
details in the section 4.
The fermions and mirror fermions are assigned to the following representations
of the gauge group SO(12) in terms of the SO(10) group decomposition
(a) The fermion sector
16T =(t ...t ,ν ,b ...b ,τ−,bc...bc,τ+, tc... tc, νc) (2)
1 3 τ 1 3 1 3 − 1 − 3 − τ L
(b) The mirror sector
16∗T =(tˆ ...tˆ ,νˆ ,ˆb ...ˆb ,τˆ−,ˆbc...ˆbc,τˆ+, tˆc... tˆc, νˆc) (3)
1 3 τ 1 3 1 3 − 1 − 3 − τ L
where the above structure form an irreducible spinor Ψ+ = (Ψ16, Ψ16∗) of the
SO(12), and the index i = 1..3 is a color index. The SO(N) gauge theories are
naturallyfreefromanomalies,the anomaliesinvolvingtheETCsectorbarecanceled
by introductionof the respective leptons andtechnileptons. The mirrorfermions in
17
our model are introduced in order to implement the Barr and Zee mechanism ,
which has been used to explain the muon-electron mass difference. In our case the
mirror fermions will replace the exotic leptons present in the Barr and Zee model.
Thefermionicassignmentsofthe16T multipletshowninEq.(2)withrespectto
the sub-group SU(3) SU(3) SU(2) SU(2) U(1) are the following
ETC ⊗ C ⊗ L ⊗ R ⊗ S
ti (3,3,2,1,1/3) , ti (3,3,1,2,1/3)
L ∼ R ∼
bi (3,3,2,1,1/3) , bi (¯3,3,1,2,1/3)
L ∼ R ∼
τ (3,1,2,1, 1) , τ (3,1,1,2, 1)
L ∼ − R ∼ −
ν (3,1,2,1, 1) , ν (¯3,1,1,2, 1).
τ τ
L ∼ − R ∼ −
The mirror fermionic assignments are similar, however mirror fermions are sin-
glets of SU(3) , as discussed in the introduction. The technifermions quantum
ETC
numbers are the same ones as the ordinary fermions. The only difference between
fermionsandtechnifermionsaretheTCquantumnumbers.Besidesthatbandtare
in different ETC representations.
In theories where the electroweak interaction is broken dynamically, the
10
fermionic mass terms arise when the ETC gauge group is broken . We shall not
speculate about the origin of this breaking. However, we will suppose that this in-
teraction and the SO(12) gauge symmetries can be broken by fundamental scalars
associated to supersymmetry or any other mechanism at the GUT scale. In the
5
reference the breaking of this GUT to SU(3) SU(2) U(1) is discussed in
C ⊗ L ⊗ Y
detail. For convenience we will reproduce the symmetry breaking pattern of this
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4 A. Doff and A. A. Natale
t^L t^R (cid:28)^L (cid:28)^R
WL WR WL WR
^bL ^bR (cid:23)^(cid:28)L (cid:23)^(cid:28)R
bL tL tR bR bL tL tR bR
(a) (b)
Fig.1. Diagramscontributingtothebottomquarkmass.Inthediagram(a)thecontributionto
thebquarkmasscomesfromthemirror-quarksofourmodelwhilethecontributionofthediagram
(b)isduetothemirror-leptons.
model
M
SO(12) a SU(4) SU(4)
−→ F ⊗ C
M
b SU(4) SU(2) SU(2) U(1)
−→ C ⊗ L ⊗ R ⊗ F
M
c SU(3) SU(2) SU(2) U(1)
−→ C ⊗ L ⊗ R ⊗ C+F
M
d SU(3) SU(2) SU(2) U(1)
−→ C ⊗ L ⊗ R ⊗ S
M
R SU(3) SU(2) U(1) . (4)
−→ C ⊗ L ⊗ Y
whereweindicateU(1) =U(1) U(1) .Themassesscales(M ,M )related
C+F C⊗ F a..d R
to the symmetry breaking of SO(12) group down to SU(3) SU(2) U(1)
C ⊗ L ⊗ Y
are determined respecting the constraints on the proton decay lifetime and the
renormalization of the weak angle bare value sin2θ to the experimental value
W
sin2θ = 0.23120(15)11. In our model the symmetry breaking of the last stage
W
shown in Eq.(4) will be implemented by a techniquark condensate characterized
by a strong interaction SU(2) symmetry at O(300)GeV. According to what we
TC
discussed above we assume that the breaking of the SU(3) SU(2) will
ETC → TC
happens near the GUT scale.
We shall concentrate only in the low energy sector (i.e. bellow 1 TeV) of our
model, SU(3) SU(2) SU(2) U(1) , and in particular in the top-bottom
C ⊗ L⊗ TC ⊗ Y
mass splitting without touching the problem of mass generation for the remaining
fermions.
3. The Top Dynamical Mass
We can decompose the fermionic content associated to the gauge group SU(3)
ETC
as shown in the following:
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A dynamical mechanism toexplain the top-bottom quark mass hierarchy 5
VETC
tL TL TR tR
Fig.2. Diagramresponsibleforthetopquarkmass.
T B
1,ω 1,ω
ψ = T , B (3,1)
3ETC 2,ω 2,ω ∼
t b
L,R L
B
1,ω
ψ = B (3¯,1) (5)
3¯ETC 2,ω ∼
b
R
where ω is a techniflavor index indicating the number(n ) of flavors necessary to
ω
produce the walking behavior for the TC group. In Eq.(5) we indicate only the
SU(3) decomposition, the SU(2) structure of (t, b) multiplet is the usual
ETC L L
one.
9
ThischoiceisinagreementwiththeschemeproposedbySikivieetal. inorder
to assure that m =m without large weak isospin violation, where we are putting
b t
6
the(LH-RH)chiralcomponentsofthetopquarkinthesameETCrepresentation,
but the bottom quark components are in different ETC representations. In this
way the dynamical top quark mass is generatedat leading order as depicted in the
Fig.(2), as happens in ordinary technicolor models.
The top quark mass is giving by the following expression:
3g2 C ΛETC d4p Σ(p)
m ETC ETC TC , (6)
t ≈ 8π2 (p2+Λ2 )(p2+Λ2 )
Z TC ETC
while atthis orderwehavem =0,andΛ is the massscaleofthe ETC bosons.
b ETC
It is useful to write the technifermion self-energy in terms of the ansatz formulated
12
in the Ref.
Σ(p) Λ2 α
TC = TC [1+bg2(Λ2 )ln(p2/Λ2 )]−γTCcos(απ) (7)
Λ p2 TC TC
TC (cid:18) (cid:19)
3C
where, in this equation, we identify γ = TC with C given by C =
TC 16π2b TC TC
1[C (R ) + C (R ) C (R )], where C (R ) (with i = 1..3) is the Casimir op-
2 2 1 2 2 − 2 3 2 i
erator for the technifermions in the representations R and R that condensate
1 2
in the representation R , and b is the coefficient of order g3 in the technicolor
3
β function. The quadratic Casimir operator for the ETC gauge group can be
TC
written as C = (N2 1)/2N = 4/3 for N = 3. Finally, the mass
ETC ETC − ETC ETC
February 2, 2008 0:40 WSPC/INSTRUCTION FILE split
6 A. Doff and A. A. Natale
scale Λ can be estimated to be of the order of the electroweak symmetry scale
TC
Λ O(300)GeV,accordingthediscussionperformedinthelastsectionthescale
TC ∼
Λ can be assumed as Λ Λ .
ETC ETC ≈ GUT
Eq.(7) interpolates between the standard operator product expansion result for
the technifermion self-energy, which is obtained when α 1, and the extreme
3 →
walking technicolor solution obtained when α 0 . As we have pointed out in
13 →
Ref. only such kind of solution is naturally capable of generating a large mass
to the third fermionic generation, which has a mass limit almost saturated by the
13
top quark mass. Moreover,as also claimed in the second paper of Ref. there are
other possible reasons to have α 0, as the existence of an IR fixed point and a
∼15 16
gluon (or technigluon) mass scale , which are closely related .
We will just consider α=0 in Eq.(7) assumingthat we canintroduce a number
of technifermions necessary to obtain the extreme walking behavior, our result for
the t quark mass (and consequently for b quark mass) will be dicussed exactly in
this case. Therefore, after its substitution into Eq.(6), with p2 =(Λ2 /Λ2 )y, we
ETC TC
verify that Eq.(6) is reduced to
3g2 C Λ Λ2 −γTC
m ETC ETC TC 1+bg2(Λ2 )ln( ETC) I(Λ2 ) (8)
t ≈ 16π2 TC Λ2 TC
(cid:20) TC (cid:21)
where we used the definition
I(Λ2 )= 1 ∞dze−zzδ−1 1 −ǫz Λ2TC dyy−ǫz . (9)
TC Γ(δ) Λ2 y+Λ2
Z0 (cid:18) TC(cid:19) Z TC
To obtain the above equation we made use of the following Mellin transform:
A −δ 1 ∞ A −ǫz
1+ǫln = dzzδ−1e−z (10)
B Γ(δ) B
(cid:20) (cid:21) Z0 (cid:18) (cid:19)
and defined ǫ=bg2(Λ2 ), with δ =γ +1. After integrationof the Eq.(9) we can
TC TC
13
show that Eq.(8) leads to
m C α Λ . (11)
t ∼ ETC ETC TC
13
Theaboveresult,whichcannaturallydescribethetopquarkmass ,resemblesthe
way the top quark acquires a mass from an elementary Higgs boson with vacuum
expectation value Λ . This behavior could be predicted based on the results of
TC
14
Ref. ,whereitisremarkedthatwhenα 0inEq.(7)thecompositeHiggssystem
∼
behaves as a fundamental one, and in this case some results obtained with the
introduction of fundamental Higgs bosons are reproduced in a dynamical scenario.
This fact underlies severalaspects of our model.
4. The bottom mass
According to what we discussed at the beginning of this work, the bottom
quark mass will be generated at 2-loop level, due to the interaction with mirror-
quarks(leptons) and the W bosons as depicted in Fig.(1). There are other pos-
L,R
sible contributions to the b quark mass, but they involve the very massive SO(12)
February 2, 2008 0:40 WSPC/INSTRUCTION FILE split
A dynamical mechanism toexplain the top-bottom quark mass hierarchy 7
gaugebosonsandarenegligiblecomparedtotheonesofFig(1).Assumingtheusual
Feynman rules, we can write the following expression for the b mass
α α d4q
m = i L R ΓµD (p q)Iβδ(l,p,q)D (p q)(q+m )Γν
b − π2 (q2 m2) µβ − δν − 6 t
Z − t
(cid:2) (cid:3)(12)
whereintheaboveequationwedefinedthefunctionIβαasthefermionloopintegral
d4l
Iβδ(l,p,q)=i Tr ΓβS (l+p q)ΓδS (l) . (13)
(2π)4 F − F
Z
(cid:2) (cid:3)
In these expressions we are introducing the following notation: Γµ..ν = γµ..νσa..d.
The σa..d, for a..d = 1,2 or 3, are SU(2) generators and the D (p q) is the
L,R µν −
W propagatorintheLandaugauge.Eq.(13),afterangularandl integration,can
L,R
be written as
m dq2 (q2+M2) q2
b f(α,n ) tˆ . (14)
m ≈− F (q2+m2)(q2+M2)(q2+M2)
t Z t L R
b
where M2 =3m2/4.In orderto obtainthis lastequation, for simplicity, we consid-
tˆ tˆ
eredthe zero momentum approximationfor the externalpropagators,we neglected
themomentumdependenceofthefermionself-energiesinsidetheloopandassumed
onlythe firstdiagramshownintheFig.(1).The massesm andm arerespectively
t tˆ
the top and mirror-topmasses, M and M are the weak vector boson masses and
L R
we introduced the function
27
f(α,n )= α α n CL CR , (15)
F 64π2 L R F 2F 2F
b b
with n indicating the number of mirrorquarks(leptons) and CL = CR = 3/4 .
F 2F 2F
Notice that if we had considered the momentum dependence in the self-energies of
b
Fig.(1) we would be lost in factors that do not bring any useful information to our
calculation. Moreover,as we argue in the sequence, even in this approximation the
calculation is finite.
SimplepowercountingshowsthattheintegralinEq.(14)isapparentlylogarith-
mically divergent, we stress two points: First, if we had considered the momentum
dependence in the self-energies of Fig.(1) the integral would be softened. Second,
and as well as important, the model that we are proposing here bears some simi-
17
larity with the one proposed by Barr and Zee some years ago , whose purpose
was the calculation of the electron mass in terms of the muon mass in the case of
elementary scalar bosons. In that model a “similar” type of logarithmic divergence
alsooccurs,whichwascanceledamongdifferentcontributionstothe electronmass;
the muon contribution and the contribution of a heavy lepton X. This cancellation
occurs when it is assumed that the muon mixes with this lepton X. In our model
we can suppose that the top and mirror top mixes at the SO(12) scale, and the
February 2, 2008 0:40 WSPC/INSTRUCTION FILE split
8 A. Doff and A. A. Natale
cancellation now will occur between the fermion and mirror fermion sector keeping
thebmassfinite.IntermsofthismixingwecanwriteEq.(14)inthefollowingform
dq2 (q2+M2) q2
m m (ω)f(α,n ) tˆ +
b ≈ − t F (q2+m2)(q2+M2)(q2+M2)
Z t L R
b dq2 (q2+M2) q2
m (ω)f(α,n ) tˆ . (16)
− tˆ F (q2+m2)(q2+M2)(q2+M2)
Z tˆ L R
b
where, for simplicity, we defined
m (ω)=m (cosω )(cosω )
t t L R
m (ω)=m (sinω )(sinω ). (17)
tˆ tˆ L R
The relation between m , m and the mixing angles ω as in the case
t tˆ L,R
17
of the Barr and Zee model can be parametrized as m (cosω )(cosω ) =
t L R
m (sinω )(sinω ). Then, after the integration of the Eq.(16), we can write the
− tˆ L R
divergent pieces of this equation as
m (Λ) m (cosω )(cosω )logΛ2 m (cosω )(cosω )logΛ2. (18)
b ∝ t L R − t L R
Now we can assume that Λ because the combination of the two terms shown
→∞
above give the finite result m (Λ)=0.
b
18
In the model of reference , which has also a common structure to the model
discussed here, the authors estimated the mixing angle of the ordinary fermions
with their mirror partners as being ω 10−3, meaning that its cosine can be
≤
approximated by 1. Therefore our model has a larger convergence than the one in
the Barr and Zee model due to the points discussed above. Now we can justify our
choice for SO(12) in order to unify the interactions at the scale M , because
GUT
this gauge group exactly accommodates one usual fermionic generation and one
mirror fermion generation necessary to keep the bottom mass finite in the present
approach.Thishappensindependentlyofthefactthatweneglectedthemomentum
dependence of the self-energies in the loops of Fig.(1).
Finally, we can evaluate the contributions to the bottom quark mass shown in
Fig.(1).Diagram(b)inFig.(1)givesacontributionidenticaltothediagram(a),and
the secondcanbeobtainedfromthe firstonejustexchanginginthecalculationthe
mass m by the mirror-lepton masses m . Taking into account both contributions
tˆ τˆ
we can write the ratio between the bottom and top quark masses as
m 243 M2 (M2 M2) (M2 M2) M2
b α α n R tˆ − R + τˆ − R log R (19)
mt ≈ 1024π2 L R F (MR2 −ML2)"(m2t −MR2) (m2t −MR2)# ML2
b
where we have already set cosω 1.
L,R ≈
It is important to note that in this model the mass scale M will be linked
R(L)
5
with the symmetry breakingscale of the SO(12) groupby the following relations
8 11α 4 M2 M2
1 sin2θ = log GUT +2log R (20)
− 3 W 6π 3 M2 M2
(cid:20) L L(cid:21)
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A dynamical mechanism toexplain the top-bottom quark mass hierarchy 9
and
α 11α M2 M2
sin2θ = log R log R . (21)
α − W 6π M2 − M2
s (cid:20) GUT L(cid:21)
In order to keep sin2θ = 0.231 and M2 1016GeV the symmetry breaking
W GUT ∝
connected to the scale M must happens at M 1013GeV. Therefore, we can
R R ∼
disregard the fermionic(mirror-fermionic) masses in comparison with the M mass
R
and put Eq.(19) into the following form
m 243 M2 M2
b α α n R log R. (22)
m ≈ 512π2 L R F (M2 M2) M2
t
R − L L
b
This last equation is independent of the mirror fermions masses introduced in the
model and as the Right-Handed symmetry is broken near at GUT scale the con-
tribution associated to this very high energy scale does not contribute to the ρ
parameter. We can regard this contribution as δρ M2/M2, therefore we do not
∝ L R
have too large corrections to the Peskin-Takeuchi T parameter, which is giving by
19
M2
αT = W 1=ρ 1=δρ. (23)
M2cos2θ − −
Z W
Inthe figurebelowweplotthe behaviorofthebottomquarkmassasafunction
of the ratio between the M and M masses.
R L
4,0
3,5
) 3,0
m (b
2,5
n^=2
2,0
n^=3
21 22 23 24 25 26 27 28
Log(MR/ML)
M2
Fig.3. Thebehaviorofthebottom quarkmass,mb(µ), asafunctionof MR2 .Intheboxweare
L
indicatingthenumber ofmirrorflavors n ≡n =2andn =3that weconsider inourmodel.
F F
Onlythestrongradiativecorrectionshavebeenconsideredtoobtainthesecurves.
b b b
February 2, 2008 0:40 WSPC/INSTRUCTION FILE split
10 A. Doff and A. A. Natale
The values of Fig.(2) correspond to the bottom quark mass at the scale µ2 =
10GeV2, assuming that the bottom quark receives strong radiative corrections at
20
low energies according to the correctionfactor given by
α 4/(11−23nf)
m (µ) m (Λ ) QCD . (24)
b ≈ b GUT α
(cid:18) GUT (cid:19)
For simplicity we considered only the QCD radiative corrections and we also
assumed α = √2G M2/π 1/21, α 1/37a, α 1/45, α (µ2 =
L F L ∼ R ∼ GUT ∼ QCD
10GeV2) 0.32 with m (µ 2M ) 178GeV and n = 6. G and α are
≈ t ≈ Z ≈ f F QCD
respectively the Fermi coupling constant and QCD coupling constant.
5. Conclusions
In this model we were able to generate a reasonable mass splitting between the
bottom and top quarks without introducing large weak isospin violation. This is
9
possible because we followed the Sikivie et al. scheme, according to which the
technicolor interaction preserves a custodial SU(2) symmetry.
V
The ratio R 1 between the bottom and top quark masses that arises in our
∼ 40
model is intimately connected to the breaking of the Right-Handed symmetry. Of
course,wearetradingtheproblemofunderstandingthelargemasssplittingbetween
the top and bottom quarks, to the more fundamental question, of understanding
why there is a verylargesplitting betweenthe Left-Rightgaugesymmetries.There
areseveralpointsinthemodelthatstillneedtobedevelopedastheintroductionof
the other families, the originof the ETC symmetry breakingas wellas of the GUT
symmetry,etc...Howeverwe expectthat the basic idea ofthe model discussedhere
may represent a different approach to solve this problem.
Acknowledgments
ThisresearchwaspartiallysupportedbytheConselhoNacionaldeDesenvolvimento
Cient´ıfico e Tecnol´ogico (CNPq) (AAN), and by Fundaca˜o de Amparo `a Pesquisa
do Estado de Sa˜o Paulo (FAPESP) (AD).
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aThis value is obtained by extrapolation of the running coupling α up to the scale M ∼
1013GeV. R R
