Supersymmetry Resurrected

Abstract

The missing standard model superpartners compel one to consider an alternative implementation for supersymmetry. The basic supermultiplet proposed here includes the photon and a charged spin 1/2 preon field. These fields are shown to yield the standard model fermions, gauge symmetries and Higgs fields. The novelty is that supersymmetry is defined for unbound preons only. Quantum group SLq(2) representations are introduced to classify topologically scalars, preons, quarks and leptons. PACS 12.60.Rc

Full text

Supersymmetry Resurrected
Risto Raitio∗
02230 Espoo, Finland
January 1, 2018
Abstract
The missing standard model superpartners compel one to consider an alternative
implementation for supersymmetry. The basic supermultiplet proposed here
includes the photon and a charged spin 1/2 preon field. These fields are shown
to yield the standard model fermions, gauge symmetries and Higgs fields. The
novelty is that supersymmetry is defined for unbound preons only. Quantum
group SLq(2) representations are introduced to classify topologically scalars,
preons, quarks and leptons.
PACS 12.60.Rc
Keywords: Preons, Supersymmetry, Quantum Groups, Standard Model
∗E-mail: risto.raitio@gmail.com
1

1 Introduction
Despite of efforts producing a vast literature on supersymmetry (SUSY) some-
thing seems to be undeniably ambiguous, if not wrong. I believe that super-
symmetry has to be implemented in a different way. I propose SUSY should
be introduced one level below the standard model (SM), namely on the quark
and lepton constituent, i.e. preon level. Then the beauty of SUSY is that a
Dirac-Maxwell supermultiplet is shown to produce the standard model fields:
quarks, leptons, gauge fields and the Higgs. The preons themselves do not par-
ticipate in strong and weak interactions [1], which are emergent phenomena in
this model.
The scheme presented in this article is based on a previous preon model of
SM particles [1,2,3]. Here I propose the updated, supersymmetric preon model
as follows
1. the elementary fields are members of a supermultiplet which includes the
photon,
2. the matter field is a light spin 1/2, charge 1/3 preon,
3. supersymmetry is valid only for preons, not for bound states of preons,
and
4. the quantum group SLq(2) is used to classify topologically scalar particles,
preons, quarks and leptons [4,5].
Graviton and gravitino must also belong to the basic supermultiplet. Gravity
is further supposed to organize the preons in bound state ‘bags’. These questions
are, however, beyond the scope of this article.
The material is organized as follows. The supersymmetric preon model is
described in section 2. Elements of supersymmetry are introduced in subsection
2.1. After the apogee, in section 3quantum group structure for scalars, preons,
quarks and leptons is described to indicate the differences and the unity of these
objects. Finally, conclusions are given in section 4. The article is aimed to be
self-contained.
2 Supersymmetric Preon Model
In case supersymmetry is familiar subsection 2.1 may be skipped by going to
subsection 2.2.
2.1 Supersymmetry
An operator Qwhich generates transformations between fermions and bosons
is an anti-commuting spinor
Q|bosoni=|fermioni,Q|fermioni=|bosoni(2.1)
2

Qand its hermitian conjugate Q†carry spin 1/2. Therefore supersymmetry
must be a space-time symmetry. The generators Qand Q†satisfy the following
algebra
{Q, Q†}=Pµ
{Q, Q}={Q†, Q†}= 0
[Pµ, Q] = [Pµ, Q†] = 0
(2.2)
where Pµis the four momentum generator of space-time translations.
Let us introduce the simplest possible supersymmetric model, the Wess-
Zumino model [6].1It contains a single chiral supermultiplet: a massless, non-
interacting left-handed two-component Weyl fermion ψand, as its superpartner,
a complex scalar field φ. The corresponding Lagrangian is
LW Z =iψ†¯σµ∂µψ−∂µφ∗∂µφ(2.3)
where ¯σ0=σ0,¯σi=−σi, i = 1,2,3. The infinitesimal supersymmetry transfor-
mation of the scalar field changes it to a fermion
δφ =ψ, δφ∗=†ψ†(2.4)
where αis an infinitesimal, anti-commuting, two-component Weyl fermion
quantity. It turns out to be constant and has the dimension of [mass]−1/2.
For the fermion the transformation is
δψα=−iσµ†α∂µφ, δψ†
˙α=iσµ˙α∂µφ∗(2.5)
The supersymmetry algebra closes only on-shell, i.e. when the classical equa-
tions of motion are valid. One would like the symmetry to hold also quantum
mechanically. For this purpose one introduces an auxiliary complex scalar field
Fwhich has no kinetic term. It has dimension [mass]2and it gives a term F∗F
in the Lagrangian. The auxiliary field is useful in connection of spontaneous
supersymmetry breaking but for the present purposes it cancels out.
In a general supersymmetric model there are one or more chiral supermul-
tiplets with both gauge and non-gauge interactions. Let us construct a general
model of masses and non-gauge interactions for particles contained in (2.3). We
want to find the most general renormalizable interactions for the Lagrangian
(2.3). Each term in the interaction Lagrangian Lint must have field content of
mass dimension ≤4. The only available terms are
Lint =−1
2Cijψiψj+CiFi+cij FiFj+c.c. −U(2.6)
where the Cij, Ci, cij and Uare polynomials in the scalar fields φi, φ∗iwith
degrees 1, 2, 0 and 4, respectively.
1A fluent introduction to supersymmetry is [7].
3

Requiring (2.6) to be invariant under supersymmetry transformations leads
to the condition that U(φi, φ∗i)must vanish. Likewise the coefficients cij are
zero, and we are left with −1/2Cijψiψj+CiFi+c.c. only. Further it turns out
that Wij cannot contain φ∗i. Therefore we have
Cij =Mij +dijk φk(2.7)
whre Mij is a symmetric mass matrix for the fermion fields, and dijk is a Yukawa
coupling of a scalar φkand two fermions ψiψjthat is totally symmetric under
interchange of i, j, k. It is convenient to write
Cij =δ2
δφiδφj
W(2.8)
where the W
W=1
2Mijφiφj+1
6dijk φiφjφk(2.9)
is called the superpotential. It is a holomorphic function of the complex scalar
fields φi. One can add a linear term Biφiin the superpotential (2.9). The
fermions and the bosons obey the same wave equation. Their masses are ob-
tained from the same mass matrix with real non-negative eigenvalues: (M2)j
i=
M∗
ikMkj .
The chiral Lagrangian density is
Lchiral =−∂µφ∗i∂µφi+iψ†i¯σµ∂µψi−1
2Wijψiψj+W∗
ijψ†iψ†j−WiW∗
i(2.10)
The U(1) gauge sector of the Lagrangian is
Lgauge =−1
4Fµν Fµν +iλ†¯
σµ∂µλ+1
2D2(2.11)
where Dis an auxiliary field to make supersymmetry consistent off-shell. The
full supersymmetric Lagrangian density is L=Lchiral +Lgauge.
Details of interactions, as well as supersymmetry breaking, are beyond the
scope of this article. I hope to return to these questions elsewhere.
2.2 Preon Model
The setup is the following. Quarks and leptons consist of three light spin 1/2,
charge zero or 1/3 particles called preons, as indicated below (2.13). At an
energy of the order 1016±1GeV quarks and leptons ionize into their constituents,
preons. Below this dividing point, I assume that the standard model, with all
its subtleties, is well behaving renormalizable theory. Above the ionization
energy, supersymmetry enters the scene: it is defined for preons, which are now
unbound.
In the simplest supersymmetric preon model one has as the basic con-
stituents the photon γand its neutral spin 1/2 superpartner, photino, denoted
4

˜m0. The second superpair is the charge 1/3, spin 1/2 preon m+and a com-
plex scalar superpartner ˜s+. All fields γ,˜m0,m+and ˜s+have two degrees of
freedom:
γ=→
←and ˜m0=↑
↓,m+=↑
↓and ˜s+
1,2(2.12)
where the horizontal and vertical arrows refer to helicity and spin, respectively,
and + and 0 refer to charge in units of 1/3 electron charge. ˜s+
1,2indicates that
there are two charged scalar fields ˜s+
1and ˜s+
2. The ˜m0is a Majorana fermion.
After small SUSY breaking the m+and ˜m0are assumed to have light mass,
of the order of the first generation quark and lepton mass scale. The ˜s+adds
two physical scalar bosons, each consisting of three ˜s+preons, with charge one.
The R-parity for fields in (2.12) is simply PR= (−1)2(spin).
The standard model is constructed as follows. In section 3it will be estab-
lished that matter originates from the j= 3/2and j= 1/2representations of
SLq(2) for fermions. Originally this was postulated from a simple analysis of
quark and lepton quantum numbers in [1].
Starting from the known quark and lepton charges it is natural to assume
the following charge quantization {0, 1/3, 2/3, 1} of which only the first (0)
and the last (1) are physical charge states. To make these charges available
with spin 1/2preons, quarks and leptons must be built as bound states of three
preons. For same charge preons fermionic permutation antisymmetry factor ijk
must be included. These arguments lead to four bound states made of preons,
which form the first generation quarks (q) and leptons (l) [1] (dropping tildes)
uk=ijk m+
im+
jm0
¯
dk=ijk m+m0
im0
j
e=ijk m−
im−
jm−
k
¯ν=ijk ¯m0
i¯m0
j¯m0
k
(2.13)
A useful feature in (2.13) with two same charge preons, on lines 1 and 2,
is that the construction provides a three-valued index for quarks on the left-
hand side, to be identified as quark SU (3) color. This is how color emerges in
the preon model. It is the way quark color was originally discovered [8]. The
corresponding gauge boson states qi¯qj→gij are in the adjoint representation
3⊗3∗=8⊕1.2The weak SU (2) left handed doublets can be read from the
first two and last two lines in (2.13): l−¯
l0→W−. The standard model (SM)
gauge structure SU (3)×SU (2) is emergent in this sense from the present preon
model. In the same way quark-lepton transitions between lines 1↔3 and 2↔4
in (2.13) are possible gauge interactions.
The horizontal lines in (2.13) can be regarded as equivalence relations of
different topological objects in the sense of section 3, and the vertical coordinate
in (2.13) giving the gauge structure of the left-hand side fields.
2Introduction of color is done slightly differently in [5].
5

The above gauge picture is supposed to hold up to the energy of about 1016±1
GeV. The electroweak interaction has the spontaneously broken symmetry phase
below an energy of the order of 100 GeV and symmetric phase above it. The
electromagnetic and weak forces take separate ways at higher energies (100
GeVE1016 GeV), the latter restores its symmetry but melts away due to
ionization of quarks and leptons into preons.This gives a cutoff for the theory.
The electromagnetic interaction, in turn, stays strong towards Planck scale,
MPl ∼1.22 ×1019 GeV. Likewise, the quark color and leptoquark interactions
suffer the same destiny as the weak force. One is left with the electromagnetic
and gravitational forces only near Planck scale.
The proton, neutron, electron and νcan be constructed of 12 preons and 12
anti-preons. The construction (2.13) is matter-antimatter symmetric on preon
level, which is desirable for early universe. The model makes it possible to
create from vacuum a universe with only matter. Corresponding antiparticles
may occur equally well, but the matter dominance case seems to have been
made. Neutral dark matter is formed of preon-antipreon pairs more likely than
ordinary matter when the temperature of the universe is lowered to a proper
free mean path value between preon collisions.
The baryon number (B) is not conserved in this model: a proton may decay
at Planck scale temperature by a preon rearrangement process into a positron
and a pion. This is expected to be independent of the details of the preon
interaction. Baryon number minus lepton number (B-L) is conserved.
The preon model is approximately conformal at energies 1016 GeV.
3 Knot Theory of Fermions
Early work on knots in physics goes back in time to 19th and 18th century [9,10].
More recently Finkelstein has proposed a model based on the quantum group
SLq(2) [4,5]. The idea stems from the fact that Lie groups are degenerate forms
of quantum groups [11]. Therefore it is of interest to study a physical theory
by replacing its Lie group by the corresponding quantum group. Finkelstein
introduced the global group SLq(2) as an extension to the SM electroweak
gauge group obtaining the group structure SU (2) ×U(1) ×SLq(2).
Knots are objects in three dimensional space. Their projections onto two
dimensional plane are considered here. Oriented knots can be characterized
by three numbers as follows. Where two dimensional curves cross there is an
overline and an underline at each point, a vertex. It has a crossing sign +1
or -1 depending on whether the overline direction is carried into the underline
direction by a counterclockwise or clockwise rotation, respectively. The sum of
all crossing signs is the writhe wwhich is a topological invariant. The number of
rotations of the tangent of the curve in going once around the knot is a second
topological invariant and it is called the rotation r. An oriented knot can be
labeled by the number of crossings N, the writhe wand rotation r. The writhe
and rotation are integers of opposite parity.
6

One can transform to quantum coordinates (j, m, m0). These indices label
the irreducible representations of Dj
mm0of the symmetry algebra of the knot,
SLq(2), by defining
j=N/2, m =w/2, m0= (r+o)/2(3.1)
This linear transformations makes half-integer representations possible. The
knot constraints require wand rto be of opposite parity, therefore ois an odd
integer.
The knot or quantum algebra for a two dimensional representation M=a b
c d 
is as follows
ab =qba bd =qdb ad −qbc = 1 bc =cb
ac =qca cd =qdc da −q1cb = 1 q1= 1/q (3.2)
with real valued q.
Any knot (N, w, r)may be labeled by DN/2
w/2,(r+o)/2(a, b, c, d). The following
expression of the algebra is associated to a (N, w, r)knot
Dj
mm0(a.b, c, d) = X
δ(na+nb,n+)
δ(nc+nd,n−)
Aj
mm0(q, na, nc)δ(na+nb, n0
+)anabnbcncdnd(3.3)
where (j, m, m0)is given by (3.1), n±=j±m,n0
±=j±m0and Aj
mm0(q, na, nc)
is given by
Aj
mm0(q, na, nc) = hn0
+i1hn0
−i1
hn+i1hn−i11/2hn+i1!
hnai1!hnbi1!
hn−i1!
hnci1!hndi1!(3.4)
where n+=na+nb,n−=nc+nd,hniq=qn−1
q−1and hi1=hiq1.
One assigns physical meaning to the Dj
mm0in (3.3) by interpreting the a, b,
c, and d as creation operators for spin 1/2 fermions. These are the four elements
of the fundamental j= 1/2representation D1/2
mm0as indicated in table 1.
m m’ preon
1/2 1/2 a/m+
1/2 -1/2 b/ ¯
m0
-1/2 1/2 c/m0
-1/2 -1/2 d/m−
Table 1.
I use in the rightmost columns of tables 1. and 2. the fermion names of both
[4] and [1].
The standard model particles are the following D3/2
mm0representations
7

m m’ particle preons
3/2 3/2 electron aaa/m+m+m+
3/2 3/2 neutrino ccc/m0m0m0
3/2 -1/2 d-quark abb/m+¯
m0¯
m0
-3/2 -1/2 u-quark cdd/m0m−m−
Table 2.
All details of the SLq(2) extended standard model are discussed in [4], in-
cluding the composite gauge and Higgs bosons and a candidate for dark matter.
I do not, however, see much advantage for introducing composite gauge bosons
in the model (gauge invariance is a local property). Therefore the model of [1]
and the knot algebra model of [4] are equivalent in the fermion sector.
The most elementary configuration is a simple loop having j= 0. These are
scalar fields. Some pairs of these loops with opposite rotation may be brought
together, e.g. in the early universe, by gravitational attraction making two
opposing j= 1/2twisted loops.
In summary, knots having odd number of crossings are fermions and knots
with even number of crossings are correspondingly bosons. Instead of consider-
ing spin 1 bound states of six preons I assumed in section 2that the SM gauge
bosons are genuine point like gauge fields. The leptons and quarks are simple
quantum knots, the quantum trefoils, with three crossings and j= 3/2. At
each crossing there is a preon. The preons are twisted loops with one crossing
and j= 1/2. The j= 0 states are simple neutral loops with zero crossings (see
figures in [4]).
4 Conclusions
The present supersymmetric preon model is an economic Dirac-Maxwell sys-
tem. Its low energy limit is the standard model. The standard model can be
constructed with the superpartners being included within the resulting model.
Compared to the early version of the preon model the new fields are the
two scalar spreons ˜shaving charge 1/3. Consequently, there must be two three
spreon scalar boson bound states with charge one in nature. Preon-antipreon
and spreon-antispreon pairs make two neutral observable scalar boson. It is
tempting to associate both the charged bound states and the neutral preon-
antipreon pairs with the Higgs bosons [12]. Even single charge 1/3 scalar bosons
may exist, which would be a clear signal for the model.
The SLq(2) group provides a solid topological basis for the fermion sector of
the present model. In [5] it was shown that the SLq(2) preon model agrees with
the Harari-Shupe (H-S) rishon model [13,14].3Unlike the said preon/rishon
authors, I do not think that (i) SM gauge bosons should be treated as bound
3The basic idea of the present model was originally conceived during the week of the ψdiscovery in November
1974 at SLAC. I proposed that the c-quark is a gravitationally excited u-quark, both consisting of three spin 1/2
and charge {0, 1/3} heavy black hole constituents. This idea met resistance. Therefore the model was not developed
further until years later.
8

states of several preons, or (ii) hypercolor is realistic for preon interactions.
In any case, a mechanism for preon binding into quarks and leptons is yet
to be developed. There are tentative models where the electron, or preon, is
considered a spinning Kerr black hole with a superconducting ‘bag’ vacuum
source [15].
It is hoped that the present preon model provides a new avenue towards
better understanding of the roles of all four interactions, and gravity in par-
ticular. In the present scenario, gravity and electromagnetism are the genuine
interactions of early cosmology. The weak and strong interactions are emergent
from the basic fermion structure of the model in (2.13). Traditional advantages
of supersymmetry (like top-stop cancellation, gauge coupling unification, etc.)
loose their meaning in their old form. Instead, a wholly new phenomenology
will have to be calculated, with the hope of including in it local supersymmetry.
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