THE ”SPIN-CHARGE-FAMILY-THEORY”, WHICH OFFERS
THE MECHANISM FOR GENERATING FAMILIES, PREDICTS
THE FOURTH FAMILY AND EXPLAINS THE ORIGIN OF
THE DARK MATTER
Norma Susana Mankoč Borštnik a
Faculty of Mathematics and Physics, University of Ljubljana, 1000 Ljubljana,
Slovenia
Abstract.The theory unifying the spin, the charges and the families predicts the
number of families, explains the origin of charges and family members, of gauge
fields and of scalar fields. It explains the contribution of the scalar fields to the
breaks of symmetries and correspondingly to properties of fermions and gauge
fields. It predicts the fourth family which might be observed at the LHC and
explains the dark matter as clusters of the stable fifth family members. All the
family members have all the charges, with the family quantum number included, in
the fundamental representations of the groups and all the vector or scalar bosons
in the adjoint representations of these groups. Massive scalar dynamical fields
(with nonzero vacuum expectation values) manifest as Yukawa fields and might
effectively be replaced at low energies as Higgs and Yukawa couplings.
1
A short presentation of the spin-charge-family-theory
A simple starting Lagrange density [1, 2] for spinors in d > (1 + 3), which
carry two kinds of the spin – the Dirac one (described by γ a , S ab = 4i (γ a γ b −
γ b γ a )) and the additional one (described by γ̃ a , with the same Clifford algebra
properties as γ a and anticommuting with γ a , S̃ ab = 4i (γ̃ a γ̃ b −γ̃ b γ̃ a ), there exists
no third kind of the spin) – no charges, and interact with only the gravitational
field through the vielbeins and the two kinds of the spin connection fields, the
gauge fields of S ab and S̃ ab ,
Lf
p0a
1
(E ψ̄ γ a p0a ψ) + h.c. ,
2
1
1
1
{pα , Ef α a }− , p0α = pα − S ab ωabα − S̃ ab ω̃abα ,(1)
= f α a p0α +
2E
2
2
=
manifests (after particular breaks of the starting symmetry) in d = (1 + 3) two
groups of four massless families. The gravitational gauge fields are assumed to
appear in the action
Z
Z
d
S=
d x E Lf +
dd x E (α R + α̃ R̃)
(2)
through the vielbeins (f α a , f α a eb α = δab ) and the spin connection fields of two
kinds (ωabc = f α c ωabα and ω̃abc = f α c ω̃abα ) as follows
R=
1 α[a βb]
1
f f
(ωabα,β − ωcaα ω c bβ ) , R̃ = f α[a f βb] (ω̃abα,β − ω̃caα ω̃ c bβ ) , (3)
2
2
a e-mail:
norma.mankoc@fmf.uni-lj.si
with f α[a f βb] = f αa f βb − f αb f βa . These gauge fields manifest in d = (1 + 3) as
the gauge fields of the observed charges and as the scalar fields which contribute
to the two successive breaks (from SO(1, 3)×SU (2)I ×SU (2)II ×U (1)II ×SU (3)
into SO(1, 3)×SU (2)I ×U (1)I ×SU (3) and further to SO(1, 3)×U (1)×SU (3)),
and determine, together with the gauge fields, the low energy properties of
fermions and bosons.
In this talk a short overview of properties of the two kinds of the scalar
dynamical fields – one (the gauge fields of S ab ) distinguishing only among the
members (α , α ∈ (u , d , ν , e)) of a family, the other (the gauge fields of S̃ ab )
only among the families – is made and the influence of scalar fields on properties
of the mass matrices of twice (decoupled) four families and on the gauge fields
presented. The reader is kindly asked to read for more information the ref. [5]
and the references cited there.
To see that the action of Eq. (1) manifests after the breaks of symmetries [2,
5, 6] all the known gauge fields, with the mass matrices included, let us rewrite
formally the action for a Weyl spinor of (Eq.(1)) as follows
X
Lf = ψ̄γ n (pn −
g A τ Ai AAi
n )ψ +
A,i
{
X
s
ψ̄γ p0s ψ} +
s=7,8
the rest,
(4)
P
where n = 0, 1, 2, 3 and τ Ai = a,b cAi ab S ab , {τ Ai , τ Bj }− = iδ AB f Aijk τ Ak .
Index A enumerates all possible spinor charges and g A is the coupling constant
to a particular gauge vector field AAi
n . Before the electroweak break stays
A = 1 for the weak charge, A = 2 for the hyper charge and A = 3 for the
colour charge. The first row of Eq.(4) manifests the dynamical
part, while
P
the second row manifests the mass term. The mass term ( s=7,8 ψ̄γ s p0s ψ),
in which the summation runs only over s = 7, 8, the operators γ s transform
quantum numbers of the right handed members into those of the left handed
partners. To the mass matrix two kinds of spin connection fields contribute
1
{pσ , Ef σ s }− , p0σ = pσ − 21 S ab ωabσ − 12 S̃ ab ω̃abσ , and we
p0s = f σ s p0σ + 2E
P
have correspondingly τ̃ Ai = a,b cAi ab S̃ ab , {τ̃ Ai , τ̃ Bj }− = iδ AB f Aijk τ̃ Ak .
The scalar fields which originate in S̃ ab and couple to only the upper four
families are responsible with their nonzero vacuum expectation values for mass
matrices of the upper four families on the tree level. The lower four families
stay massless up to the electroweak break, when the scalar fields which are orthogonal to the ones which determine mass matrices of the upper four families,
gain nonzero vacuum expectation values. To the mass matrices of the lower
four families also the scalar fields originating in ωabs contribute.
While are the mass matrices of different members of a family on the tree
level strongly correlated (u has the same of diagonal matrix elements as ν and
d has the same of diagonal matrix elements as e ), the so far evaluated one loop
corrections [3] give a real hope that the loop corrections to all orders take care
of the great differences in the properties of the family members for the lower
four families. For neutrinos a Majorana like term appears, which might be to
high extent responsible for the smallness of masses of the lower three families.
The influence of the loop corrections on the upper four families is expected to
be much smaller than on the lower four families, since the scalar fields originating in the ”Dirac kind” of the spin connection fields (ωss0 s” b ), which distinguish
among the family members (not among families c ), gain nonzero vacuum expectation values at the electroweak break, while the upper four families gain
masses at for several orders of magnitude higher scale d .
The upper group of four families (Σ = II) do not couple to the lower one
(Σ = I). Correspondingly the mass matrices of each family member (α ∈
(u , d , ν , e)) demonstrate twice four by diagonal matrices
Mα =
M α II
0
0
MαI
,
(5)
α II,I
where M α II,I have on the tree level (M(o)
) the structure

M(o)
−a1
 b
=
 0
c
b
−a2
c
0
0
c
a1
b

c
0 
.
b 
a2
(6)
Accordingly there are two stable families of quarks and leptons. The first one
(u1 , d1 , ν1 , e1 ) and the fifth family one (u5 , d5 , ν5 , e5 ). All the fifth family
members are expected to have quite comparable masses due to the fact that
the main dependence on the family members quantum number appears at the
electroweak break, as explained above. It follows accordingly that the lightest
baryon is very likely the neutron (n5 ) e .
b The
scalar fields originating in the ”Dirac kind” of spin γ s manifest after the elec0
troweak break in terms presented in the last line of Eq. (8) as e Q As + g 1 cos θ1 Q0 ZsQ +
0
g 2 cos θ2 Y 0 AY
s .
c The gauge scalar fields of τ̃ Ai , which distinguish among families, are presented in Eq. (8)
~ Ã
~ ÑR + g̃ Ỹ 0 Ỹ 0 ÃỸ 0 + √
g̃ 2
in the second and the third line as {g̃ ÑR Ñ
(τ̃ 2+ Ã2+ + τ̃ 2− Ã2− ) +
R
ÑL
0
1
s
s
2
s
s
~ Ã
~
g̃
1+ Ã1+ + τ̃ 1− Ã1− ).
+ g̃ Q̃ Q̃0 ÃQ̃
g̃ ÑL Ñ
L s
s + √2 (τ̃
s
s
d Together with the upper four families also the vector gauge fields, which are the super~ 2 and A4 , gain masses when SU (2)II × U (1)II symmetry breaks into U (1)I .
position of A
m
m
e Properties of the fifth family members are under consideration [3, 4].
0
2
Mass matrices on tree and beyond tree level
After the electroweak break the Lagrange density for spinors (Eq. (1)) is equal
to
Lf
=
p0m
=
+
ψ̄ (γ m p0m − M )ψ ,
g1
Q0
1+
1−
pm − {e Q Am + g 1 cos θ1 Q0 Zm
+ √ (τ 1+ Wm
+ τ 1− Wm
)+
2
0
g2
2− 2−
Am ) ,
(7)
g 2 cos θ2 Y 0 AYm + √ (τ 2+ A2+
m +τ
2
with the mass term
ψ̄ M ψ
p0s
= ψ̄ γ s p0s ψ ,
g̃ 2 2+ 2+
~ Ã
~ ÑR + g̃ Ỹ 0 Ỹ 0 ÃỸ 0 + √
(τ̃ Ãs + τ̃ 2− Ã2−
= ps − {g̃ ÑR Ñ
R s
s
s )
2
g̃ 1 1+ 1+
~ Ã
~ ÑL + g̃ Q̃0 Q̃0 ÃQ̃0 + √
(τ̃ Ãs + τ̃ 1− Ã1−
+ g̃ ÑL Ñ
L s
s )
s
2
0
0
+ e Q As + g 1 cos θ1 Q0 ZsQ + g 2 cos θ2 Y 0 AYs } , s ∈ {7, 8} .
(8)
We have before the two SU (2) × U (1) breaks two times (Σ ∈ (II, I)) four
α
massless vectors ψΣ(L,R)
for each member of a family α. Let i, i ∈ (1, 2, 3, 4),
denotes one of the four family members of each of the two groups of massless
families
α
α
α
α
α
ψΣ(L,R)
= (ψΣ
(9)
1 , ψΣ 2 , ψΣ 3 , ψΣ 4 )(L,R) .
Let Ψα
Σ(L,R) be the final massive four vectors for each of the two groups of
families, with all loop corrections included,
α
ψΣ
(L,R)
=
VΣα Ψα
Σ (L,R) ,
VΣα
=
VΣα(o) VΣα(1) · · · VΣα(k) · · · .
(10)
Then we have for the mass term of Eq. (7)
α
0
αΣ
α
α
0 α†
αΣ α
< ψΣ
|ψΣ
VΣ |Ψα
L |γ M
R >=< ΨΣ L |γ VΣ M
ΣR > .
(11)
Rough analyses (without loop corrections included) of the properties for the
lower group of four families [2,6] and for the fifth family members [7] were done
by taking into account the so far observed properties of family members f and
of the dark matter observations and can be found in the cited references. The
f The calculations of the lower four families properties were done under the assumption
that loop corrections change the off diagonal matrix elements while keeping approximately
the mass matrices symmetry of Eq. (6).
prediction was made that the fourth family members can be possibly observed
at the LHC or at somewhat higher energies.
The analyses with the loop corrections taken into account are in preparations [3]
manifesting so far their strong dependence on family member quantum number.
This gives correspondently a real hope that the loop correction will change the
tree level mass matrices (which have the off diagonal matrix elements of u equal
to those of ν and equivalently for d and e) in agreement with the observations.
3
Dark matter candidates
The spin-charge-family-theory predicts a stable fifth family, with masses above
the fourth family masses, let us say > 10 TeV, and pretty much below 1013
GeV, which is a possible scale of the first of the two SU (2) × U (1) breaks.
We followed [7] the history of the stable fifth family members in the expanding
universe from the time when they start to decouple from the cosmic plasma,
through the freezing out procedure and the colour phase transition, up to the
today’s matter, showing that the fifth family members, the (colourless) baryons
and anti-baryons and neutrinos and anti-neutrinos are very likely the dark
matter constituents. Neutrinos and antineutrinos are expected to annihilate
during the electroweak phase transition to the density which is in agreement
with cosmological observations of the dark matter density. Let me point out
that the estimations need to be followed by more accurate calculations, at least
to clarify several assumptions.
We calculated the today number density of baryons and anti-baryons of the
fifth family members (nc5 (T0 )) as a function of the fifth family members mass.
Taking into account the estimation inaccuracy the interval for the fifth family
masses followed
10 TeV < m5 c2 < 4 · 102 TeV.
(12)
We used the Bohr-like model to estimate the cross section for the fifth family
neutrons π(rc5 )2
10−8 fm2 < σc5 < 10−6 fm2 .
(13)
One can see that the cross section of the fifth family ”nuclear force” is at least
10−6 × smaller than the cross section of the first family nuclear force. This
explains why is the dark matter so inert.
We studied [7] also properties of the fifth family baryons and anti-baryons
and neutrinos and anti-neutrinos in direct measurements done so far.
My prediction is, after several discussions with the members of most of the
groups performing direct measurements, that if DAMA/LIBRA measures our
fifth family clusters, other direct measurements will confirm the DAMA/LIBRA
results.
4
Comments on standard model and extensions of it as seen from
spin-charge-family-theory
Let us try to understand the standard model as an effective approach of the
spin-charge-family-theory. The main ”extension” of the proposed theory is in
offering the mechanism for generating families, and in the unification of all the
internal degrees of freedom. The inclusion of the right handed neutrino as a
regular family member, which is a necessary part of the spin-charge-familytheory, is a very small extension of the standard model.
Mass matrices of fermions of the lower four families are in the spin-chargefamily-theory, according to Eq.(8), determined by the scalar fields through the
operator
1
78
0
~ Ã
~ ÑL
Q
1~ 1 ~
Q0 0 Q
Y0 0 Y0
Φ̂I∓ =(∓) {g̃ ÑL Ñ
L ∓ + g̃ τ̃ Ã∓ + eQA∓ + g Q Z∓ + g Y A∓ },
(14)
78
78
with (∓)= 12 (γ 7 ∓γ 8 ). One can easily check (see ref. [1,5]) that the operator (−)
transforms the weak and the hyper charge of the right handed uR and νR (−)
78
into those of the left handed ones, while (+) does the same for the dR and eR .
All the scalar bosons as also all the vector bosons of Eq. (14) have all the charges
78
in the adjoint representations of the charge groups. The operator (∓) obviously
does what in the standard model the Higgs by ”dressing” the right handed
quarks and leptons with the appropriate charges does. The part of the operator
~ Ã
~ ÑL +g̃ 1 ~τ̃ 1 Ã
~ 1 +e Q AQ +g Q0 Q0 Z Q0 +g Y 0 Y 0 AY 0 },
of Eq. (14), that is {g̃ ÑL Ñ
L
∓
∓
∓
∓
∓
takes care of the mass matrices on the tree level, and together with the vector
massive fields also beyond the tree level.
The spin-charge-family-theory evidently manifests that the scalar fields are new
interactions, which the Yukawa couplings of the standard model already do.
78
If replacing the operator (∓) with the vacuum expectation value of an ”effective Higgs” ΦvI
∓ , which is a weak doublet of appropriate hyper charge like
it is a Higgs of the standard model (see the ref. [5]) and taking into account
g1
1
0 Q0
1− 1−
[τ 1+ A1+
Am ] + g 1 sin θ1 Q AQ
that p0m = pm − √
m + g cos θ1 Q Am ,
m +τ
2
(Eq. (7)) one finds that
(p0m Φ̂I∓ )† (p0m Φ̂I∓ ) = {
0
(g 1 )2 1+ 1− m
g1
Q0 m
vI
Am A
+(
)2 AQ
}T r(ΦvI†
mA
∓ Φ∓ ). (15)
2
2 cos θ1
2
v
vI
Assuming, like in the standard model, that T r(ΦvI†
∓ Φ∓ ) = 2 , we extract from
the masses of gauge bosons one information about the vacuum expectation
values of the scalar fields, their coupling constants and their masses.
We see that the standard model Higgs and the Yukawa couplings are in the
spin-charge-family-theory represented by the bunch of scalar fields, which only
effectively manifest as the Higgs and the Yukawa couplings, but are expected
to show up at the LHC as a bunch of scalar fields.
Let me at the end comment on models [10], which extend the standard model
by assuming that the observed families manifest the SU (3) flavour group and
that the scalar fields are in the ”bi-fundamental” representations of the SU (3)
flavour group. From the point of view of the spin-charge-family-theory with the
scalar fields in the adjoint representations with respect to all the groups look
”bi-fundamental” representations of the SU (3) flavour group quite artificial. It
is not easy to connect the spin-charge-family-theory scalar fields to scalar fields
in the ”bi-fundamental” representations of the SU (n) group.
5
Concluding remarks
The spin-charge-family-theory [1,2,5,6] is offering the way beyond the standard
model by proposing the mechanism for generating families of quarks and leptons and consequently predicting the number of families at low energies and
the mass matrices for each of the family member. It explains the appearance
of charges as well.
It predicts the fourth family to be possibly measured at the LHC or at some
higher energies and the fifth family which is, since it is decoupled in the mixing matrices from the lower four families and it is correspondingly stable, the
candidate to form the dark matter [7].
The proposed theory predicts that there are two kinds of scalar fields, which
interact with fermions through the Dirac spin and through the second kind of
the Clifford operators (anticommuting with the Dirac ones). The Dirac one
distinguishes among the family members, the second one among the families.
Beyond the tree level these two kinds of scalar fields and the vector massive
fields start to contribute coherently, leading hopefully to the measured properties of the so far observed three families of fermions and to the observed weak
gauge fields.
At low energies these scalar dynamical fields manifest effectively pretty much,
although they all have the charges in the adjoint representations of the groups,
as the standard model Higgs and Yukawa couplings together. It is expected,
however, that the differences will show up at least when searching for these
scalar fields.
It is obvious from the point of view of the spin-charge-family-theory that the
scalar fields represent new kind of interactions, what manifest already the standard model Yukawas. The proposed theory predicts also that there is no supersymmetric partners of existing fermions or bosons, at least not below the
unification scale of the three known interactions.
References
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http://arxiv.org/abs/1005.2288.
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