Low-energy sector of 8-dimensional General Relativity: Electro-Weak model and neutrino mass

0.19 MB·English

by  Francesco Cianfrani

Low-energy se
tor of 8-dimensional General
Relativity: Ele
tro-Weak model and neutrino
mass
1 123
Fran
es
o Cianfrani , Giovanni Montani
1
ICRA(cid:22)International Center for Relativisti
Astrophysi
s
Dipartimento di Fisi
a (G9), Università di Roma, (cid:16)Sapienza",
Piazzale Aldo Moro 5, 00185 Rome, Italy.
8
0
2
0
ENEA C.R. Fras
ati (Dipartimento F.P.N.), Via Enri
o Fermi 45, 00044 Fras
ati,
2
Rome, Italy.
n
a 3
J
ICRANet C. C. Pes
ara, Piazzale della Repubbli
a, 10, 65100 Pes
ara, Italy.

8
1
e-mail: montanii
ra.it

2
fran
es
o.
ianfranii
ra.it
v
0
4
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6
0
6 PACS: 11.15.-q, 04.50.+h
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/
c
q
-
r
g
v: Abstra
t
i
X V4 S1 S3
In a Kaluza-Klein spa
e-time ⊗ ⊗ , we demonstrate that the dimensional re-
r SU(2)
a du
tion of spinors provides a 4-(cid:28)eld, whose asso
iated gauge
onne
tions are
geometrized. However, additional and gauge-violating terms arise, but they are highly
β
suppressed by a fa
tor , whi
h (cid:28)xes the amount of the spinor dependen
e on extra-

oordinates. The appli
ationof this framework to the Ele
tro-Weak model is performed,
β
thus giving a lower bound for from the request of the ele
tri

harge
onservation.
Moreover, we emphasize that also the Higgs se
tor
an be reprodu
ed, but neutrino
masses are predi
ted and the (cid:28)ne-tuning on the Higgs parameters
an be explained, too.
1 Introdu
tion
One of the main issues of modern Physi
s is to re
ast all intera
tions into a uni(cid:28)ed pi
-
ture. This aim requires to (cid:28)x a new paradigm of our knowledge, able to explain the deep
di(cid:27)eren
e between gravity, geometrized by General Relativity, and strong-ele
tro-weak
intera
tions, having the form of gauge theories.
We have two di(cid:27)erent ways of a

omplishing the above purpose: to re
over even gravity
as a gauge theory (an example is Poin
aré Gauge Theory [1℄) or to sear
h for a geomet-
ri
al formulation of the Standard Model. Sin
e a gauge theory for gravity seems to work
only in a quantum level [2℄[3℄, then a uni(cid:28)
ation pi
ture based on a geometri
al point of
view is suitable for a
lassi
al formulation.
In this paper, we
onsider a Kaluza-Klein (KK) approa
h for the geometrization of
the Ele
tro-Weak model. This s
heme [4℄ [5℄ (for a review see [6℄, [7℄ or [8℄) pursues
the Einstein's idea to give a physi
al
ontent to all the metri
degrees of freedom; in
parti
ular, it is based on re
ognizing our four-dimensional spa
e-time as embedded in
a multidimensional one, whose additional (o(cid:27)-diagonal) metri

omponents determine
gauge bosons. To re
on
ile this s
heme with our four-dimensional phenomenology, the
extra-spa
e is assumed to be
ompa
ti(cid:28)ed at distan
es forbidden to experiments. There-
fore, our all-day Physi
s is a low-energy approximation with respe
t to
ompa
ti(cid:28)
ation
energy s
ales.
This framework was proved to be very useful for bosons, sin
e Yang-Mills Lagrangian

omes out from the dimensional redu
tion of the Einstein-Hilbert a
tion. However the
introdu
tion of (cid:28)elds able to reprodu
e (gauge
oupled) four-dimensional fermions is a
mu
h more di(cid:30)
ult task, ex
ept for the Abelian
ase. In fa
t, by expanding alá Fourier
U(1)
fun
tions along the (cid:28)fth dimension, transformations
an simply be reprodu
ed by
x5
-translations [8℄.
With the aim of dealing with the Ele
tro-Weak model, here we propose a phenomeno-
SU(2)
logi
al approa
h for a gauge intera
tion: spinors are introdu
ed with an extra-

oordinates dependen
e whi
h is indu
ed by their low-energy
hara
ter. In this respe
t,
S3
we require that spinors satisfy the Dira
equation when averaged on the extra-spa
e ,
thus taking into a

ount the un-observability of this spa
e. Then, we
onsider an ex-
β−1
pansion in an order parameter ,
hara
terizing the dependen
e on extra-
oordinates.
SU(2)
The geometrization of gauge
onne
tions is easily a

omplished this way at the
β−1
lowest order; moreover, the (cid:28)rst additionalterms, of order , implyviolationsof gauge
β
symmetries. Therefore, an estimate of the parameter , on
e this framework is applied
to the Ele
tro-Weak model and
ompared with experimental limits, is obtained.
Furthermore, we introdu
e the multidimensional analogous of the Higgs (cid:28)eld, by whi
h
we
an give masses to all parti
les, in
luding neutrinos; then, its extra-dependen
e gives
φ†φ
a term having a
oe(cid:30)
ient of the
ompa
ti(cid:28)
ation length order. This term
an
a

ount for the (cid:28)ne tuning required to stabilize the Higgs mass [9℄. Reprodu
ing the
spontaneous symmetry breaking me
hanism, we also obtain a non-vanishing photon
β
mass, from whi
h a lower bound on is obtained.
In parti
ular, the organization of all this topi
s in the paper is the following: in se
tion
2 we analyze the standard way spinors are introdu
ed in a KK framework; in se
tion 3
2
β−1
we turn to our phenomenologi
al approa
h: we determine, at the lowest order in , a
S3 SU(2)
solution of the Dira
equation on , for whi
h in se
tion 4 the geometrization of
gauge
onne
tions is performed; in se
tion 5 we
onsider the appli
ation of previous
U(1)
results to the geometrization of the Ele
tro-Weak model, where the hyper
harge
β−1
symmetry has to be in
luded; in se
tion 6 non-standard
ouplings,
oming from
terms, are analyzed; in se
tion 7 we reprodu
e the Higgs me
hanism, with the new fea-
β
ture of a neutrino mass and a
onstraint on due to the predi
tion of a massive photon
too; (cid:28)nally, in se
tion 8 brief
on
luding remarks follow.
2 Spinors within the Kaluza-Klein framework
The development of the original KK theory [4℄ [5℄ provided a framework where the
geometrizationof the ele
tro-magneti
(cid:28)eld
ould be performed. After intera
tionsother
than gravity were re
ognized as the Yang-Mills ones, in
reasing interest in su
h models
appeared. In fa
t, enlarging the dimensions of spa
e-time via a
ompa
t homogeneous
manifold, the boson se
tor of gauge theories
an be re
overed from the Einstein-Hilbert
a
tion [6℄, [7℄.
Problems arise when fermions are introdu
ed, be
ause they are treated as matter (cid:28)elds
and relevant short
omings of the model take pla
e when the following point of view
is addressed. In a straightforward approa
h, multidimensional spinors are provided by
simply extending the four-dimensional formalism; this line of thinking has to fa
e non
trivial questions, su
h as the fermion mass, the
hirality problem and so on.
In fa
t, eigenvalues of the Dira
equation on the extra-spa
e behave, under the KK
hypothesis, like masses and they are of the
ompa
ti(cid:28)
ation s
ale order; this fa
t leads
to the model in
onsisten
y be
ause, for the multidimensionalspinors, no zero-eigenvalue
state exists [10℄. There are two main ways to avoid this result: the introdu
tion of
torsion [11℄ or of non-geometri
al gauge (cid:28)elds [12℄. We stress that these possibilities
look rather ad-ho
and the last one
on(cid:29)i
ts with the spirit of KK models, sin
e it
introdu
es non-geometri
al bosons.
As far as the
hirality is
on
erned, when the Ele
tro-Weak model is
onsidered in KK,
di(cid:27)erent transformation properties for left-handed and right-handed spinors have to be
implemented in a geometri
al way. Therefore, the hope was that the right-handed and
left-handed zero modes of the Dira
operator behaved di(cid:27)erently under n-bein rotations.
Thispossibilityisruledoutby the Atiyah-Hirzebru
h theorem[13℄, despite the
ase non-
geometri
algauge(cid:28)eldsarepresent. Thisway, theKKprogram
anbeappliedtospinors
at the pri
e of introdu
ing external gauge bosons [14℄, thus, a

ording to this point of
view, the most important result (the geometrization of the boson
omponent) would be
destroyed.
However, as said above, these results are based on the assumption that multidimensional
and four-dimensional spinors
oin
ide. Here we propose a di(cid:27)erent approa
h, where 4-
spinors are the reli
of the full (cid:28)eld, after the dimensionalredu
tion has been preformed.
In this sense, we develop a phenomenologi
almodel, able to reprodu
e the Ele
tro-Weak
theory from standard KK hypothesis.
3
3 Dira
equation on the 3-sphere
Sin
e in a KK framework intera
tions other than gravity are geometrized (by virtue
of o(cid:27)-diagonal metri

omponents), the properties of matter (cid:28)elds are (cid:28)xed by their
dependen
e on extra-
oordinates.
The aim of this se
tion is to state the form of spinorial (cid:28)elds, on a spa
e-time manifold
with a
ompa
ti(cid:28)ed three-dimensional sphere.
SO(1;3) SO(3)
The reli
un-broken symmetry group in the full tangent spa
e is ⊗ and
therefore we
an build up an eight-
omponent representation in the following way:
Ψ = χ ψ r,s = 1,2
r rs s
(1)
χ SU(2) ψ
s
being a representation and Dira
spinors.
Hen
e, weassumetofa
torizethedependen
eonfour-dimensionalandextra-dimensional
χ = χ(y) ψ = ψ(x)

oordinates (denoted by x and y respe
tively), su
h that .
A physi
al spinor (cid:28)eld should arise as the solution of the
orresponding Dira
equation,
whi
h takes the following form in su
h a KK manifold
χγ(µ)(eµ ∂ Γ )ψ +γ(µ)em (∂ χ)ψ +γ(m)[(em ∂ Γ )χ]ψ = 0
(µ) µ − (µ) (µ) m (m) m − (m) (2)
where Greek and Latin letters indi
ate four-dimensional and extra-dimensional
ompo-
nents, respe
tively, while indi
es in parenthesis are n-bein ones. For extra-dimensional
γ γ(m) = γ σ σ
5 (m) (m)
matri
es we have , being Pauli matri
es.
In equation (2) we also assume a non-Riemannian spa
e-time manifold, by taking as
spinorial
onne
tions just those adapted to the four-dimensional and to the extra-
dimensional spa
e, respe
tively i.e.
Γ =4Γ
(µ) (µ) ;
(cid:26) Γ =3Γ = iσ (3)
(m) (m) 2 (m)
su
h a
hoi
e is a natural
onsequen
e of breaking the Lorentz symmetry into the dire
t
SO(1;3) SO(3)
produ
t of two di(cid:27)erent group, and .
Now, the e(cid:27)e
tive four-dimensional theory for Dira
spinors is obtained after the
om-
χ
putation of terms.
ψ χ
To avoid mass terms of the
ompa
ti(cid:28)
ation order for , we should require to be a
solution of the massless Dira
equation on the 3-sphere. However, no exa
t solution of
the massless Dira
equation on the three sphere (and in general on a
ompa
t mani-
fold [10℄) exists. However, phenomenologi
ally, we do not need so mu
h; in fa
t, when
we
onsider the redu
tion of a multidimensional theory, the un-observability of extra-
S3
dimensions has to be taken into a

ount. The dynami
s on being undete
table, we
have to integrate on su
h a spa
e to perform the
orre
t splitting [15℄, i.e. we must take
an average pro
edure on the extra-
oordinates dependen
e, whi
h, for the extra-spa
e
homogeneity,
orresponds to a unit weight.
Thus, we look for a solution of the following averaged Dira
equation
i
d3y√γγ(m)(em ∂ σ )χ = 0.
Z (m) m − 2 (m) (4)
S3
4
χ
In previous works [8℄[15℄[16℄, we
onsidered the following form for
χ = 1 e−2iσ(p)rsλ((pq))Θ(q)(ym)
r
√V (5)
V S3 λ
with the volume of and a
onstant matrix satisfying
1
(λ−1)(p) = √ γem ∂ Θ(p)d3y.
(q) V Z − (q) m (6)
S3
Θ
Be
ause ofthearbitrarinessofthe fun
tions, we impli
itlyassumed the
ommutativity
χ λ−1
between and its partial derivatives. However, this request implies to have a
χ ∂ χ
m
vanishing determinant. In fa
t, if and
ommute, it
an be shown by some algebra
Θ(q) = f(y)c(q) c(q)
that , 's being
onstants, and therefore we
ould get
1
(λ−1)(p) = c(p) √ γem ∂ fd3y = c(p)d
(q) V Z − (q) m (q) (7)
S3
whose determinant vanishes.
c(p)
Here we allow to have a dependen
e on y-
oordinates, instead, but we
ontrol the

ommutativity by virtue of an order parameter, i.e. we
an require the validity of the
pi
ture (5) to an arbitrary degree of approximation. Hen
e, we write
1
Θ(p) = c(p)e−βη η > 0
β (8)
∂ c(p) c(p)∂ η
m (m)
and, without loss of generality, we require that ∼ , so that
i
∂ χ = β σ χλ(p)c(q)∂ η +O(β−1) .
m 2 (cid:20) (p) (q) m (cid:21) (9)
χ
By substituting the last expression into equation (4), and by expanding in a series of
β
the parameter , we (cid:28)nd
i
d3y√γγ(m)em ∂ χ = σ χ+O(β−1)
Z (m) m 2 (m) (10)
S3
β
thus, the bigger the parameter is, the better the spinor approximates the solution of
the massless Dira
equation. We stress that, in the above mentioned previous works, we
χ
had to negle
t to re
over the spinor (5) as a solution of the Dira
equation.
β
Moreover, sin
e determines the size of the phase, it is related to the extra-dimensional
momentum; buttheamountofenergy needed toex
iteextra-dimensionalmodesisofthe
order of the
ompa
ti(cid:28)
ation s
ale, so that, in the low energy limit, a weak dependen
e
β
on extra-
oordinates (i.e. a big value for ) is expe
ted.
Finally, it is
lear that we are performing a low energy e(cid:27)e
tive theory, sin
e the spinor
is no longer a good approximate solution as we approa
h the
ompa
ti(cid:28)
ation energy
s
ales. Therefore, we do not need to dis
uss the renormalizability of our model.
5
SU(2)
4 Geometrization of the gauge
onne
tion
In a Kaluza-Klein s
heme we
an geometrize boson degrees of freedom, but, unless we
introdu
e Supersymmetries, fermions have to be treated as matter (cid:28)elds. So, we
an
state that the geometrization of intera
tions is performed only after re
overing gauge

onne
tions starting from free spinors.
V4 S3
Let us assume KK hypothesis for the spa
e-time manifold ⊗ , i.e.
(µ) (ν)
g = η e e
µν (µ)(ν) µ ν
 e(m) = αW(m)
µ µ
 ,
 em(µ) = 0 (11)
(m) (n)
γ = η e e
 mn (m)(n) m n


g γ 
µν mn
and being the four-dimensional and the extra-dimensional metri
, respe
tively,
α W(m) em
with givingthe lengthof the extra-spa
e, while µ are gauge bosons and (m) 3-bein
S3
ve
tors on whi
h are Killing ve
tors of the 3-sphere (for their expli
it form see [17℄).
em SU(2)
Thus, the set { (m)} satis(cid:28)es the algebra of the group
1
en ∂ em en ∂ em = C(p) em ,
(n) n (m) − (m) n (n) α (n)(m) (p) (12)
(p)
C
with (n)(m) stru
ture
onstants.
Let us
onsider the free Dira
a
tion
i~c
S = [D Ψ¯γ(A)Ψ Ψ¯γ(A)D Ψ]√g√γd4xd3y
(A) (A)
2 Z − (13)
V4⊗S3
together with the form of the spinor found in the previous se
tion; after the dimensional
β−1
redu
tion, we obtain at the lowest order in the right gauge
oupling, i.e.
i~c i
S = ψ¯ γ(µ) D eµ W(m)σ ψ c.c.+O(β−1) √ gd4x.
2 Z (cid:20) (cid:18) (µ) − 2α (µ) µ (m)(cid:19) − (cid:21) −
This way, we showed that it is possible to geometrize in the low-energy limit a SU(2)
gauge theory.
β
However, a
lear indi
ation of additional higher order terms in (and not removable by
the manifold symmetries) appears; the request to disregard these terms will give a lower
β
bound for the parameter in the appli
ation to the Standard Model.
Now, we
an say that the e(cid:27)e
t of su
h terms is to break the gauge invarian
e by adding
an intera
tion of the following type
i
...+ W(m)M(n)ψ¯σ ψ +...
2β µ (m) (n) (14)
(n)
M
(m) being
1 1
M(n)σ = d3yχ† dsχsσ λ(r)e−ηβem ∂ c(s)χ1−s
(m) (n) V Z Z (r) (s) (m) m (15)
S3 0
a
onstant matrix with no obvious properties.
6
5 Geometrization of the Ele
tro-Weak Model
We now apply results of previous se
tions to the Ele
tro-Weak Model.
V4 S1 S3
Firstofall,letus
onsiderthespa
e-timemanifold ⊗ ⊗ ,wherethegeometrization
SU(2) U(1)
of a ⊗ gauge theory
an be performed in a KK ba
kground [16℄. In this
(m)
W
µ
sense, we assume the well-known form for the metri
tensor, where gauge bosons
B
µ
and arise as o(cid:27)-diagonal
omponents
g +B B +δ W(m)W(n) α′B αe W(m)
µν µ ν (m)(n) µ ν µ m(m) µ
 
j = α′B 1 0 .
AB  ν 
  (16)
 
 (n) 
 αen(n)Wν 0 γmn 
 
α′ S1
with giving the length of the spa
e.
The main point is that by the dimensional redu
tion of the Einstein-Hilbert a
tion
c4
S = nR jd4xdy0d3y
−16πG Z − (17)
(n) V4⊗S1⊗S3 p
(m)
B W
µ µ
the Yang-Mills Lagrangian density for gauge bosons and
omes out
c3 1 1
S = √ g R B B gµρgνσ δ F(n)F(m)gµρgνσ +R ,
−16πG Z − (cid:20) − 4 µν ρσ − 4 (n)(m) µν ρσ 1+3(cid:21)
V4
R
1+3
being the
urvature of the extra-dimensional spa
e. By the Standard Weinberg
rotation, Z and photon (cid:28)elds arise.
On
e spinors have been introdu
ed, we a

ount for isospin degrees of freedom by a
S3
dependen
e on
oordinates of the form (5). Instead, as usual in KK theories, the
U(1)
geometrization of a gauge
onne
tion is easily obtained by a phase dependen
e on
S1
the variable.
Finally, we are able to reprodu
e any quark generation and fermion family of the Stan-
dard Model by the following multidimensional spinors
y0 n
Ψ (x;y;θ) = einsθχ (y)ψ (x) Ψ (x;y;θ) = einrθψ (x); θ = Y = r
Lr rs Ls Rr Rr α′ r 6
(18)
Y ψ ψ ψ
r r R L
being the hyper
harge of the four-dimensional spinor , while and stand for
γ
5
the two four-dimensional
hirality ( ) eigenstates. In the present phenomenologi
al
approa
h, the
hirality problem is over
ome be
ause we
an deal dire
tly with four-
dimensional (cid:28)elds; su
h 4-spinors are the reli
of multidimensional states
hara
terized
by a di(cid:27)erent dependen
e on extra-dimensions (a

ounting for the di(cid:27)erent isospin num-
bers).
ψ
L
Now, for left-handed (cid:28)elds we have to identify with isospin doublets. Therefore, it is
ψ
R
a natural
hoi
e to think of the two
omponents of as right-handed partners of ea
h
7
doublet.
Sin
e we developed our model in an eight-dimensional spa
e-time, we deal with multi-
dimensional spinors having 16
omponents. We suggest to re
ast these
omponents, so
that any spinor
ontains a quark generation and a fermion family, i.e.
einuLθu einuRθu
χ L R
1  (cid:18) eindLθd (cid:19)  1  (cid:18) eindRθd (cid:19) 
Ψ = L Ψ = R .
L √Vα′  χ einνLθνeL  R √Vα′  einνRθνeR  (19)
 (cid:18) eineLθe (cid:19)   (cid:18) eineRθe (cid:19) 
L R
   
This way, from 6 multidimensionalspinors, one for ea
h familyor generation and one for
ea
h
hiralityeigenstate, we areabletoreprodu
e allStandardModelparti
les. Further-
more, in doing that, we give an explanation for the equal number of quark generations
and fermion families.
Chirality feature of the model The assignment of the above dependen
e for left-
handed and right-handed (cid:28)elds
ould look ad-ho
to reprodu
e quantum numbers of
Standard Model parti
les, but an explanation for it
omes from the massless
hara
ter
of the involved (cid:28)elds. In fa
t, sin
e masses are going to be generated by an Higgs-like
me
hanism, then the evolution of right-handed and left-handed (cid:28)elds is totally un
orre-
lated in a 4-dimensionalba
kground, before the spontaneous symmetry breaking to take
pla
e. This
on
lusion is modi(cid:28)ed by terms of the Dira
equation due to the presen
e of
the extra-spa
e. This
an be re
ognized from their expli
it form, whi
h reads as follows
(for the representation of Dira
matri
es see [16℄)
γ Ψ Ψ
σ(m)D 5 1 D 1
 (m)(cid:18) γ Ψ (cid:19)  0 I  (7)(cid:18) Ψ (cid:19) 
γ(m)D Ψ+γ(7)D Ψ = 5 2 + 2
(m) (7) γ Ψ (cid:18) I 0 (cid:19) Ψ
 σ(m)D 5 3   D 3 
 (m)(cid:18) γ Ψ (cid:19)   (7)(cid:18) Ψ (cid:19) 
5 4 4
   
(20)
and
learly indi
ates that starting from left-handed (cid:28)elds, mixing terms with right-
handed ones are present in the equation giving the dynami
s.
Despite the proper de(cid:28)nition of
hirality in a multi-dimensional s
enario, an indepen-
dent dynami
s for right-handed and left-handed (cid:28)elds
an be reprodu
ed by treating
4-dimensional
hirality eigenstates like fundamental (cid:28)elds. In this respe
t, to assume
a di(cid:27)erent dependen
e on extra-
oordinates is then required in order to restore the
proper 4-dimensional phenomenology, i.e. the
hirality of the SU(2) gauge intera
tion.
Furthermore, it also avoids the emergen
e of a
orrelation between the two 4-
hirality
eigenstates and, at the same time, the appearan
e of huge Kaluza-Klein mass terms.
In fa
t, in order to reprodu
e proper quantum numbers, one ends up with terms of the
form Ψ
1
Ψ¯γ(7)Ψ = (Ψ¯ Ψ¯ ) (Ψ¯ Ψ¯ ) 0 I  (cid:18) Ψ2 (cid:19) 
1 2 3 4 (cid:18) I 0 (cid:19) Ψ3 (21)
(cid:0) (cid:1)  
 (cid:18) Ψ (cid:19) 
4
 
8
Ψ
whi
h vanishes, at least
lassi
ally, as far as the (cid:28)eld is a
hirality eigenstate.
All these spe
ulations
learly indi
ate that a deep
onne
tion exists between the avoid-
an
e of Kaluza-Klein mass terms and the
hirality issue.
From the results of previous se
tions, one
an easily re
ognize that starting from the
Dira
Lagrangian density for ea
h spinor
i~c
S = [Ψ¯ γ(A)D Ψ +Ψ¯ γ(A)D Ψ +c.c.]√ γgd4xdθd3y
Ψ L (A) L R (A) R
− 2 Z − (22)
V4⊗S1⊗S3
we obtain,bydimensionalredu
tion, ane(cid:27)e
tivea
tion
ontainingthegaugeintera
tion,
β−1
plus additional terms of order, i.e.
i~c i i
S = [ψ¯ γ(µ)(D + eµ W(m)σ + y eµ B )ψ +ψ¯ γ(µ)(D +
Ψ − 2 Z L (µ) 2α (µ) µ (m) 2α′ L (µ) µ L R1 (µ)
V4
i i
+ y eµ B )ψ +ψ¯ γ(µ)(D + y eµ B )ψ +c.c.+O(β−1)]√ γgd4x
2α′ R1 (µ) µ R1 R2 (µ) 2α′ R2 (µ) µ R2 − (23)
Hen
e, we have just performed the geometrization of the Ele
tro-Weak intera
tion for
spinors.
Then, we introdu
e
oupling
onstants by rede(cid:28)ning gauge bosons
B kg′B W(i) kgW(i) (i = 1,2,3)
µ ⇒ µ µ ⇒ µ (24)
and, by imposing the Lagrangian density to
oin
ide with the Ele
tro-Weak model one,
the following relations between
oupling
onstants and extra-dimensions lengths arise
~ 2 ~ 2
α2 = 16πG = 0.18 10−31cm α′2 = 16πG = 0.33 10−31cm.
(cid:18)gc(cid:19) × (cid:18)g′c(cid:19) × (25)
Be
ause of these estimates, less than two order of magnitudegreater than Plan
k length,
we expe
t we'll be able to explain the stabilization of the extra-spa
e, in a quantum
gravity framework (for a
lassi
al me
hanism of stabilization see [18℄).
6 Non standard
ouplings
In se
tion 2 we have found a solution of the Dira
equation on the three-sphere at
β−1
the lowest order in and we have a
hieved the geometrization of the Ele
tro-Weak
intera
tion. At higher orders, there is no
lear indi
ationthat the geometrizationof
on-
ne
tions
an be performed by virtue of the Dira
equation solution, therefore we expe
t
β−1 SU(2)
some deviations (of order ) to o

ur on the
omponent of the Ele
tro-Weak
model.
In parti
ular, these additional terms are of the form (14) and they introdu
e new ver-
β
texes, whi
h break gauge symmetries. We
an get an estimate of the parameter by

onsidering that these vertexes will indu
e modi(cid:28)
ation on
ross se
tions. These mod-
β−1
i(cid:28)
ations are of the order for Standard Model pro
esses, be
ause of interferen
es,
9
O(β−2)
while probabilities for forbidden de
ays will be . This fa
t enables us to estimate
β
a lower bound for from
urrent limits on forbidden de
ays, sin
e
ontributions to pre-

ision tests, like
orre
tions to partial widths of W and Z, will result to be far below the
experimental un
ertainty.
Let us
onsider the partial width for the de
ay of a neutron in a proton plus a
ouple of
neutrino-antineutrino, whi
h a
tually has the following experimental limit [19℄
Γ(n p+ν +ν¯ )/Γ < 8 10−27.
e e tot
→ ∗ (26)
Sin
e su
h a de
ay breaks the ele
tri

harge
onservation, its Feynman diagrams must

ontain at least one vertex responsible for the violation, i.e. of the form (14).
(cid:1)
Fig. 1
For instan
e, a possible Feynman diagram is in (cid:28)gure 1, where we have a vertex with
W3
µ
Z and a
ouple of quarks u-d; by substituting into the photon and the Z (cid:28)elds, we
obtain, among the others, a term des
ribing this
oupling in the expression (14) , i.e.
i
¯ (1) (2)
...+ cosθ Z ψ(M σ +M σ )ψ +...
2β W µ (3) (1) (3) (2) (27)
β−1
Therefore, the amplitude for the pro
ess will be of order of , the
ross se
tion being
β−2
O( ) and, this way, we get
1
10−27 β 1014.
β2 ≤ ⇒ ≥ (28)
β
With su
h a big value for ,
orre
tions to the
ross se
tions of the allowed Standard
Model pro
esses,
oming out from interferen
es with new vertexes, are suppressed far-
below the present experimental un
ertainty for re
ent ele
troweak pre
ision tests, too.
β−1 10−14
Forinstan
e, the partialwidthsof Zand Wre
eive
orre
tionsof the ≤ , whi
h
(10−4)
is far below the experimental un
ertainty (whi
h is at most 0 ).
7 The Higgs me
hanism
In order to give masses to parti
les, we have to introdu
e a (cid:28)eld, whose dimensional
redu
tion gives the Higgs boson.
Let us
onsider the following form for su
h a multi-s
alar (cid:28)eld
Φ 1 e−3iθφ
Φ = 1 = χ 1
(cid:18) Φ2 (cid:19) √Vα′ (cid:18) e3iθφ2 (cid:19) (29)
10