††thanks: Contribution of an agency of the U.S. government; not subject to
copyright

# Advanced code-division multiplexers for superconducting detector arrays

K. D. Irwin<EMAIL_ADDRESS>H. M. Cho  W. B. Doriese  J. W. Fowler  G. C.
Hilton  M. D. Niemack  C. D. Reintsema  D. R. Schmidt  J. N. Ullom  L. R. Vale
National Institute of Standards and Technology, Boulder, CO 80305

###### Abstract

Multiplexers based on the modulation of superconducting quantum interference
devices are now regularly used in multi-kilopixel arrays of superconducting
detectors for astrophysics, cosmology, and materials analysis. Over the next
decade, much larger arrays will be needed. These larger arrays require new
modulation techniques and compact multiplexer elements that fit within each
pixel. We present a new in-focal-plane code-division multiplexer that provides
multiplexing elements with the required scalability. This code-division
multiplexer uses compact lithographic modulation elements that simultaneously
multiplex both signal outputs and superconducting transition-edge sensor (TES)
detector bias voltages. It eliminates the shunt resistor used to voltage bias
TES detectors, greatly reduces power dissipation, allows different dc bias
voltages for each TES, and makes all elements sufficiently compact to fit
inside the detector pixel area. These in-focal-plane code-division
multiplexers can be combined with multi-gigahertz readout based on
superconducting microresonators to scale to even larger arrays.

##  I Introduction

Arrays of superconducting transition-edge sensors1 (TES) are widely used to
detect millimeter-wave, submillimeter, and x-ray signals. The development of
kilopixel arrays has required cryogenic signal multiplexing techniques. To
date, all deployed arrays use either time-division multiplexing (TDM)2 or
frequency-division multiplexing (FDM)3. The potential advantages of
multiplexing TES devices with Walsh codes have been anticipated4, 5, and code-
division multiplexing (CDM) circuits are now emerging6 that can significantly
increase the number of pixels multiplexed in each output channel, with more
compact modulation elements.

Code-division multiplexing (CDM) shares many of the advantages of both TDM and
FDM. In CDM, the signals from all TESs are summed with different Walsh-matrix
polarity patterns. In the simplest case of two-channel CDM, the sum of the
signals from TESs 1 and 2 would first be measured, followed by their
difference. The original signals can be reconstructed from the reverse
process. CDM can use the same room-temperature electronics as TDM. Unlike TDM,
CDM does not suffer from the aliasing of SQUID noise by $\sqrt{N}$, where $N$
is the number of pixels multiplexed. CDM uses smaller filter elements and
simpler room-temperature electronics than FDM, and it allows dc biasing of the
TES sensor, making it easier to achieve optimal energy resolution7.

Several CDM circuits have been demonstrated. One implementation, flux-summed
CDM6 ($\rm{\Phi}$-CDM), has been used to achieve average multiplexed energy
resolution of 2.78 eV $\pm$ 0.04 eV FWHM at 6 keV with a small array of TES
x-ray microcalorimeters8. Here we present a more advanced CDM multiplexing
circuit topology that allows scaling to much larger multiplexing factors than
$\rm{\Phi}$-CDM. In current-summed CDM (I-CDM), only one SQUID amplifier is
required for each column of detectors. The current from all TES calorimeters
in a column flows out in one pair of wires, with a coupling polarity that is
switched at each pixel by compact, ultra-low-power switches in the focal plane
itself. These switches can be placed underneath an overhanging x-ray absorber,
so that separate wires need not be extracted from each pixel. In I-CDM, the
voltage bias source for the TES calorimeters does not dissipate power at the
cold stage, making the power dissipation in even megapixel arrays manageable.
Because the dc voltage bias source for the TES calorimeters is naturally
multiplexed, different bias voltages can be chosen for each pixel. Finally,
the number of address wires required scales only logarithmically with the
number of rows multiplexed. Logarithmic encoding of the address lines is made
possible by the periodic nature of the response of the superconducting
interferometer switches to address flux6.

##  II Superconducting polarity modulation switches

I-CDM requires a circuit that can modulate the polarity of the current
coupling from a TES to its SQUID amplifier. The modulation has unity gain;
amplification occurs only after the signal from many TES pixels is summed with
different polarities. After the signal bandwidth is limited by a one-pole
$L/R$ low-pass filter formed by a Nyquist inductor $L_{\rm{nyq}}$ and the TES
resistance, superconducting switches steer the current from the TES into one
of two pathways that couple to the SQUID with opposite polarity. Because the
polarity modulation occurs at much higher frequency than the bandwidth of the
signal, there is no degradation in performance from detector noise aliasing.
We have already fabricated and tested appopriate polarity modulators9 as part
of a previous CDM circuit implementation. Fig. 1a shows the current coupled to
the SQUID ($I_{\rm{squid}}$) vs. the input current ($I_{\rm{in}}$) for the two
settings of the modulator.

Figure 1: Polarity modulation. (a) Experimental measurements of a polarity
modulator. Current into the SQUID ($I_{\rm{squid}}$) vs. the input current
($I_{\rm{}_{in}}$) in two different states: positive (solid, positive slope)
data are for no applied address flux; negative (dashed, negative slope) data
are for address flux $\Phi_{\rm{add}}=\Phi_{0}/2$. Data are shown for three
different pixels summed into one SQUID (black, red, and green — the data for
all three curves are nearly identical). The inflection points near $\pm
7\rm{\mu A}$ are indicative of the transition to the normal state above the
current-carrying capacity of the modulator used in this experiment. (b) A
photograph of the new generation of polarity-modulation switches that are used
in I-CDM. The switch contains four Josephson junctions. Junction 1 and 2 are
separated by a serial gradiometer, as are junctions 3 and 4. Junctions 2 and 3
are adjacent, and behave as a single junction with twice the critical current.
This design provides larger operating margins and higher current-carrying
capacity. An expanded view is shown for the serial gradiometer separating
junctions 3 and 4.

The polarity modulator contains superconducting switches10, 11 that are based
on compact, low-inductance SQUIDs controlled by an applied flux. These
switches are designed so that their critical currents modulate from a maximum
value at zero flux (zero applied address current) to very near to zero at a
flux of $\Phi_{0}/2$. At zero applied address flux, the switch is closed, and
the TES current flows in parallel through its Josephson junctions, which are
in a zero-voltage state. At a flux of $\Phi_{0}/2$, the combined current flow
through the parallel Josephson junctions in each switch drops near zero, and
the switch acts as a normal resistor with a value orders of magnitude larger
than the TES resistance. The ‘open’ resistance is large enough that it
introduces no significant Johnson-Nyquist current noise. The switches used in
I-CDM consist of four Josephson junctions rather than two (Fig. 1b), which
allows both larger operating margins and higher current-carrying capacity than
two-junction switches10. The new switches work well, have high yield, and have
wide operating margins.

##  III Current-Summed (I-CDM) Array Architecture

The I-CDM array architecture presented here uses the polarity modulators shown
in Fig. 1. Figure 2a shows the I-CDM circuit diagram for a small 4-pixel
array. In this circuit, each TES is wired in series with a Nyquist inductor
that is large enough that the current through the TES is approximately
constant during a multiplexed frame. Each TES (and its Nyquist inductor) is
coupled to a polarity modulator, schematically represented in the figure as
two single-pole double-throw (SPDT) switches that connect the two electrodes
of the TES and Nyquist inductor alternately to the two wires coming from the
SQUID coil. The current from all four TESs is summed with different polarities
into one pair of wires, with the polarity of each summation determined by the
state of the associated pair of SPDT switches. The column is read out with a
single SQUID amplifier on the same silicon chip.

Figure 2: The I-CDM multiplexer. (a) A schematic of a four-pixel I-CDM
multiplexer. The currents through all TES pixels (variable resistors in the
figure) and their Nyquist inductors $L_{\rm{nyq}}$ are modulated in a Walsh
pattern and summed in parallel into the input coil of a SQUID. In the example
state shown in the schematic, TES pixels 1 and 4 are coupled to the SQUID with
positive polarity, while TES pixels 2 and 3 are coupled with negative
polarity. A voltage bias is applied to the detectors by means of a current
$I_{\rm{bias}}$, which is injected into the primary of a coupled inductor.
$I_{\rm{bias}}$ induces a voltage $U(t)$ on the secondary of the inductor. (b)
A periodic bias current ramp $I_{\rm{bias}}$ chosen to bias TES 1 at zero
voltage, and TES 2-4 at $\approx$ 38 nV. (c) The induced voltage levels $U(t)$
on the secondary. (d) The induced voltage $V(t)$ on the series combination of
TES 3 and its Nyquist inductor $L_{\rm{nyq}}$ (e) The voltage across TES 3,
which is approximately constant except for a 0.16 % rms ripple.

In I-CDM, the TES detectors are dc biased. The average voltage bias level
$\overline{V}$ on each pixel is set by applying a repeating linear current
ramp $I_{\rm{bias}}$(t) to the coupled inductor in Fig. 2a. One example of an
$I_{\rm{bias}}$(t) pattern is shown in Fig. 2b. The current ramp
$I_{\rm{bias}}$(t) induces a repeating series of voltage levels $U(t)$ on the
secondary of the coupled coil (Fig. 2c). The polarity of the coupling between
each pixel and the bias signal $U(t)$ is modulated in a Walsh code. Figure 2d
shows the modulated voltage bias across the series combination of TES 3 and
its Nyquist inductor, $V(\rm{TES3}+L_{\rm{nyq}})$, for pixel 3. In the four-
pixel example in Fig. 2, the vector of average voltages $\overline{V}$ seen by
the four TES pixels is $\overline{V}_{i}=\sum W_{ij}U_{j}/4$ (summed over
j=1..4), where $W_{ij}$ is the 4$\times$4 Walsh matrix and $U_{j}$ is the
value of the voltage $U(t)$ for each pixel during the four Walsh periods. The
voltage across the TES itself stays approximately constant because the
impedance of $L_{\rm{nyq}}$ is large compared to the TES resistance at the
modulation frequency. The voltage across pixel 3, V(TES3), is shown as an
example in Fig. 2e.

All Walsh matrices are non-singular, thus any combination of average TES bias
voltages $\overline{V}$ can be established by multiplying the desired values
of $\overline{V}$ by the inverse of the Walsh matrix,
$W_{ij}^{-1}=(1/4)W_{ij}$. The first TES is typically disconnected because its
output is not modulated in the Walsh code, so it doesn’t share the same
benefits from nulling common-mode pickup. For large multiplex factors, this
results in only a small loss in the number of pixels. Thus, we set
$\overline{V_{1}}=0$, which has the added benefit that $\rm{I_{bias}}$ will
return to the same level after each frame. The four voltages on the secondary
$U_{i}$ must be $U_{i}=\sum W_{ij}\overline{V}_{i}$ (summed over j=1..4), or

$\vspace{-3 pt}\begin{pmatrix}U_{1}\\\ U_{2}\\\ U_{3}\\\ U_{4}\\\
\end{pmatrix}=\begin{pmatrix}1&1&1&1\\\ 1&1&-1&-1\\\ 1&-1&-1&1\\\ 1&-1&1&-1\\\
\end{pmatrix}\begin{pmatrix}0\\\ \overline{V_{2}}\\\ \overline{V_{3}}\\\
\overline{V_{4}}\\\ \end{pmatrix}.$ (1)

The example of Fig. 2 is for the case in which all of the TES voltages are
chosen to have the same value $V_{0}$. In this case, $U_{1}=3V_{0}$, and
$U_{2}=U_{3}=U_{4}=-V_{0}$. The numerical values chosen in Fig. 2 are for a
particular TES detector design, with $V_{0}$=38 nV, row periods of 250 ns, and
$L_{nyq}$=100 nH. In this example, the voltage across the TES is approximately
constant (Fig. 2e) with an rms ripple of only 0.16 %. Since this ripple occurs
over periods much shorter than the response time of the TES, it does not
measurably degrade the achievable energy resolution.

As the polarity of each TES is switched, it generates a back-action voltage.
This would first appear to be a source of crosstalk, since it also acts on
other TES pixels. However, over a full frame, the crosstalk back-action is
null due to the orthogonality of the Walsh vectors. The back-action of the
switching on the TES itself, averaging over multiple frames, appears as an
additional source of resistance $R_{\rm{s}}=L_{\rm{sw}}/\tau_{\rm{dwell}}$ in
series with the voltage bias 6, where $\tau_{\rm{dwell}}$ is the average time
between switching. The source resistance, $R_{\rm{s}}$, must be kept small
compared to the TES bias resistance $R_{0}$ to maintain a voltage bias.
Another constraint is placed by Josephson-frequency oscillations: the voltage
applied to the ‘open’ switch will cause a small ac leakage current to
oscillate at the frequency $V/\Phi_{0}$. The switch circuit must be designed
so that the Josephson oscillations are out of band.

Figure 3: (a) A photograph of part of a 32$\times$32 TES x-ray calorimeter
array that was fabricated as a geometric test. This array uses Mo-Cu TES
thermometers, and will later be integrated with x-ray absorbers cantilevered
over the multiplexer components. Inset: a magnified view of four pixels. The
area of relieved silicon nitride membranes is the darker outline around the
‘I’-shaped Cu x-ray absorber attachment structures. (b) A full lithographic
layout of the I-CDM multiplexer, which fits beneath overhanging x-ray
absorbers. Each two-lobed blue coil is a Nyquist inductor. The symmetry of the
lobes ensures that the magnetic field on the adjacent TESs is close to zero.
The polarity switches are the circuit elements running horizontally between
the I-shaped posts.

I-CDM has great potential to increase the scalability of both TES bolometer
instruments for far-infrared / millimeter-wave measurements, and for TES x-ray
detectors. In order to demonstrate the potential to incorporate I-CDM
multiplexer components within an x-ray detector pixel, we have fabricated a
32$\times$32 TES x-ray calorimeter test array with pixels on a 300 $\mu$m
pitch, and room for the I-CDM multiplexer components (Fig. 3a). In a full
implementation, x-ray absorbers will be cantilevered over the multiplexer
components, and connected to the ‘I’-shaped absorber attachment structures. We
have also developed a full lithographic layout of an I-CDM multiplexer
integrated in this test array (Fig. 3b).

##  IV Conclusions

I-CDM uses compact multiplexing elements that fit beneath an x-ray absorber in
a TES array with a 300 $\mu$m pixel pitch. I-CDM modulation elements are much
smaller than the LC filters used in FDM and the microwave resonators used in
MKIDs and microwave SQUIDs. I-CDM does not use shunt resistors to voltage bias
TES detectors, greatly reducing the power dissipation, and making it possible
to scale to larger arrays.

The output bandwidth provided by dc SQUID amplifiers is typically a few
megahertz, which limits the total multiplexing factor. Greater output
bandwidth and much higher multiplexing factors can be achieved by using
microwave SQUID multiplexers12 as the readout SQUIDs in I-CDM instead of
traditional dc SQUIDs.

###### Acknowledgements.

We acknowledge support from NASA under grant NNG09WF27I.

## References

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  * 3 J. Yoon, J. Clarke, J.M. Gildemeister, A.T. Lee, M.J. Myers, P.L. Richards, and J.T. Skidmore, Appl. Phys. Lett. 78, 371, (2001).
  * 4 B. Karasik, and W. McGrath, Proc. of 12th Int’l Symp. on Space Terahertz Technology, 436, (2001).
  * 5 M. Podt, J. Weenink, J. Flokstra, and H. Rogalla, Physica C 368, 218, (2002).
  * 6 K.D. Irwin, M.D. Niemack, J. Beyer, H.M. Cho, W.B. Doriese, G.C. Hilton, C.D. Reintsema, D.R. Schmidt, J.N. Ullom, and L.R. Vale, Supercond. Sci. Technol. 23, 034004, (2010).
  * 7 L. Gottardi, J. van der Kuur, P.A.J. de Korte, R. Den Hartog, B. Dirks, M. Popescu, and H.F.C. Hoevers, AIP Conference Proceedings 1185, 538, (2009).
  * 8 J.W. Fowler, W.B. Doriese, G.C. Hilton, K.D. Irwin, D.R. Schmidt, G. Stiehl, J.N. Ullom, and L.R. Vale, Proc. of 14th Int’l Workshop on Low Temp. Detectors, Heidelberg, Germany, Aug. 1-5, 2011, submitted.
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# Quantum construction of a unitary SU(2/1) model of the electro-weak
interactions with 2 Higgs doublets.

Jean Thierry-Mieg
NCBI. NLM. NIH Bldg 38A,
8600 Rockville Pike,
Bethesda, MD 20894, USA.
Tel 1 (301) 435 49 21 Fax 1 (301) 480 92 41 E-mail:
and
Laboratoire de Physique Théorique et d’Astroparticules,
CNRS, Montpellier, France<EMAIL_ADDRESS>
###### Abstract:

The interactions and even the number of the Higgs scalar fields are not fixed
in the $SU(2)U(1)$ standard model of the electro-weak interactions and the
intrinsically chiral nature of the weak interactions is not explained.
Embedding $SU(2)U(1)$ into the Lie superalgebra $SU(2/1)$ fills these gaps.
The 2 smallest representations of $SU(2/1)$ adequately describe the electron,
neutrino, up and down quarks and correlate their chiralities with their $U(1)$
charges, and the Higgs fields have the quantum numbers of the odd generators.
But so far, there was an apparent conflict with unitarity, because the quark
representation is not Hermitian and the super-Killing metric is not positive
definite. We solve this paradox by assuming the existence of 2 complex Higgs
doublets minimally coupled to the Fermions via the chiral projections of the
odd generators of $SU(2/1)$. We find that Lagrangian induced by the Fermion
loops is unitary, thanks to the balance between the leptons and the quarks
needed to cancel the triangle anomaly and that the super-Jacobi identity
guarantees that the photon remains massless after symmetry breaking. In
addition, the Lagrangian has a classical geometric interpretation in terms of
the curvature of the corresponding Hermitian algebra. Assuming that the
relative strength of the scalar and vector couplings does not depend on the
number of families constrains the mass of the Higgs to
$M^{2}_{H^{0}_{1}}+M^{2}_{H^{0}_{2}}=32/9\;M^{2}_{W}=2\;(107.2GeV)^{2}$.
Contrary to grand-unified or Wess-Zumino supersymmetric models, the $SU(2/1)$
internal superunification does not predict any unobserved particle besides the
2 Higgs doublets.

SU(2/1), superalgebra , chiral Yang Mills , renormalization, Hermitian
algebra, super symmetry, Higgs mass , standard model , chirality , neutrino ,
quark

††preprint: PTA/07-058

##  1 Background

The Weinberg-Salam $SU(2).U(1)$ model of the electro-magnetic and weak
interactions [Wei67, Sa68] is extremely accurate. At high energy, the Fermions
are described as massless $SU(2)$ doublets of left spinors: the electron
neutrino left doublet $(e_{L},\nu_{L})$ with $U(1)$ charge (-1), and the up
and down left quark doublet $(u_{L},d_{L})$ with charge (2/3), accompanied by
right singlets: the right electron $(e_{R})$ with charge (-2) and the right
quarks $(u_{R}),(d_{R})$ with respective charges (4/3) and (-1/3). The right
neutrino is $SU(2)$ and $U(1)$ neutral and drops out from the model.

The corresponding $SU(2)$ and $U(1)$ Yang-Mills gauge fields [YM54] are the
$(W^{+},W^{-})$, the $Z^{0}$ and the photon. The existence and interactions of
the $W^{+}$, the $W^{-}$ and the $Z^{0}$ were beautifully confirmed over the
last 30 years. The only component of the model that has not yet been observed
is the scalar Higgs field. In the minimal approach, it is predicted to be a
scalar $SU(2)$ doublet $\phi$ with $U(1)$ charge (-1). The Higgs potential is
assumed to be of the form $V(\phi)=(\phi^{2}-v^{2})^{2}$ So, in the vacuum the
Higgs field does not vanish. The gauge group breaks down to the electro-
magnetic $U(1)$, the maximal subgroup leaving $v$ invariant. The photon and
the left neutrino remain massless. All other particles acquire a mass
proportional to $v$ (see for example [AL73]).

This construction is superb, but the number of arbitrary parameters is very
large. Since the left and right chiral states are independent, we need to
choose 10 charges to describe the leptons and the quarks. For example, a
massless electrically charged particle, which experimentally does not exist,
would be acceptable at the classical level and is only ruled out by the study
of the quantum anomalies. Also, since the Higgs field plays such a central
role, one would wish to derive it from first principle. The Yang-Mills gauge
fields are well understood, but the Higgs potential and even the number of
Higgs fields needed in the standard model remain arbitrary.

In 1979, Ne’eman [N79] and Fairlie [F79] have independently proposed to
consider the $SU(2/1)$ Lie superalgebra as a basis for describing the weak
interactions. Indeed, the fundamental representation, recalled in section 3,
exactly corresponds to the lepton triplet if we grade it by chirality. The
representation is Hermitian (3.2) and scaling the trace of all the even
matrices to a common value seemed to predict a electro-weak angle
$sin^{2}\theta_{W}=1/4$, a good value at the time, although higher than
today’s experimental value $.22$. In addition, the odd generators have the
quantum numbers of the Higgs fields. The quarks were either left out [F79] or
mentioned as a counterargument [N79]. But remarkably, it was found a few weeks
later [DJ79, NT80] that $SU(2/1)$ also admits a 4 dimensional representation,
recalled in section 4, exactly fitting the quarks, with 2 right singlets and a
left doublet with electric charges $2/3$ and $-1/3$.

These results, summarized in a recent Physics Report [NSF05], raised some
interest but also some doubts [S92]. A superalgebraic structure seems in
direct conflict with unitarity. The supertrace, which is the natural invariant
of a superalgebra, yields a negative sign for the propagator of the $U(1)$
vector, the quark representation (4.3) is neither Hermitian nor anti-
Hermitian, and even the Hermitian nature of the lepton representation is
artefactual, since the odd generators of antilepton representation are anti-
Hermitian (3.8). It was therefore often argued that the $SU(2/1)$ superalgebra
could not play a role in the quantum field theory. The present paper reverses
the situation.

##  2 Results

By following literally the paradigm of the minimal couplings and turning
immediately to the renormalization theory, we predict 2 Higgs doublets $H$ and
$K$. This solves the problem of the metric and allows to adjust freely the
electro-weak angle. We prove that in the symmetry-breaking vacuum the photon
remains massless because of the super-Jacobi identity. For a particular choice
of the scalar coupling constants, we find a scalar Ward identity insuring that
the 1-loop counterterms do not depend on the number of families and derive the
mass relation
$M^{2}_{H^{0}_{1}}+M^{2}_{H^{0}_{2}}=32/9\;M^{2}_{W}=2\;(107.2\;GeV)^{2}$. The
whole construction plays on balance between the leptons and the quarks which
ensures the cancellation of the triangle anomaly [BIM72]. The theory is
however incomplete, because the scalar loops and the gluon loops disturb the
superalgebraic structure.

We start in section 5 from the assumption that the adjoint representation of
the $SU(2/1)$ superalgebra directly describes the Bosons of the theory, and
that the quark and lepton representations, graded by chirality, describe their
interactions to the Fermions. The even generators are gauged as usual as Yang-
Mills vector fields. But since the odd generators are not Hermitian, we
propose (5.3) to separate the left and right interactions and construct 2 sets
of scalar fields, $\Phi_{L}$ and $\Phi_{R}$ by projecting out the relevant
chiral part of the odd matrices, and to define the interactions
${\overline{\Psi}_{L}}\;\Phi_{L}\;\Psi_{R}+{\overline{\Psi}_{R}}\;\Phi_{R}\;\Psi_{L}$.
We then construct the rest of the Lagrangian by studying the counterterms
induced by the Fermion loops.

As shown in (5.4), the left-trace of the odd matrices induces the propagator
of the scalar fields. It is antisymmetric and proportional to the super-
Killing metric, as befits a superalgebra. But since the $\Phi_{L}$ and
$\Phi_{R}$ are distinct, the propagator does not vanish by symmetrization and
can be rediagonalized (5.9) in the form of 2 Higgs scalar complex doublets,
$H$ and $K$. $H$ is coupled to the negatively charged right particles
(electron and down quark) and $K$ to the positively charged right up quark.
This re-shuffling depends on the same balance between the leptons and the
quarks which cancels the triangle anomaly and the equations are similar. In
the later case, one computes the triple vector Fermion loop, which is possibly
spoiled by the impossibility to regularize the chiral Fermions in a gauge
invariant way, and depends on a supertrace condition (4.5). In the present
paper we compute the 2 scalars and 4 scalars Fermion loops, which are a priori
spoiled by the non Hermitian nature of the $SU(2/1)$ scalar-Fermion couplings.
In all cases, we find that the respective contributions of the leptons and the
quarks cancel each other and induce a canonical counterterm.

In section 6, we construct the Higgs potential. A priori, since we have 2
Higgs doublets, there are 3 possible $SU(2)U(1)$ invariant quartic terms
$V_{1}=(H^{2})^{2}+(K^{2})^{2}$ which controls the mass of the neutral Higgs
fields, a kind of vector product squared $V_{2}(H,K)$ which controls the mass
of the charged Higgs fields, and a scalar product squared $V_{3}=(H.K)^{2}$.
We show that $V_{2}$ vanishes if $H$ and $K$ are parallel as a consequence of
the super-Jacobi identity, and that $V_{3}$ is absent, because of the lepton
quark compensation. The potential can then be rewritten in terms of a simple
quadratic form $\widetilde{G}$ (6.2) which plays the role of the Killing
metric and contains a free parameter $\alpha$ later associated (10.3) to the
electro-weak angle. This construction gives the structure of the scalar
potential, but not the strength of the couplings.

In section 7, we show that for a particular value of the scalar Fermion
coupling, we have a Ward identity insuring that the Fermion loops do not
modify the relative strength of the scalar and vector couplings and that the
scalar potential renormalizes like $g^{2}$. This indicates that the mass of
the neutral Higgs fields should satisfy the relation
$M^{2}_{H^{0}_{1}}+M^{2}_{H^{0}_{2}}=32/9\;M^{2}_{W}=(151.6\;GeV)^{2}$. In the
symmetric situation, we would have $M_{H^{0}_{1}}=M_{H^{0}_{2}}=107.2\;GeV$.
However, we also observe that other quantum corrections disrupt the
universality of the scalar-Fermion coupling, indicating that the theory is
still incomplete.

In section 9, we show that the chiral decomposition of the Higgs scalars, in
conjunction with the Dirac 1-forms $\Gamma$ of [TM06], naturally constructs a
covariant differential corresponding to the SU(2/1) Hermitian algebra first
defined by Sternberg and Wolf [SW78]. The chirality operator and the complex
Higgs structure conspire so that the corresponding curvature 2-forms is well
defined and indeed valued in the adjoint representation of the $SU(2)U(1)$
even Lie algebra.

In section 10, we use the quadratic form (6.2) and the 2 CP conjugated
curvatures (9.12-13) to construct the vector Lagrangian and show that the
square of the H-algebra curvature tensor reproduces the Higgs quartic
potential induced by renormalization. Within this $SU(2/1)$ framework, the
electro-weak angle is free.

The theory is not complete and several important problems are presented in the
discussion. But even considering these difficulties, the results so far are
new and unexpected. Close to 30 years after the initial proposal [N79, F79],
this is the first time that $SU(2/1)$ is shown to play a role in the quantized
theory, and the first time that the super-Killing metric is used, both in the
construction of the propagators (5.9 and 10.1) and of the potential (6.2,6.4),
while explicitly respecting unitarity.

##  3 The $SU(2/1)$ lepton representation

The smallest nontrivial simple Lie superalgebra is $SU(2/1)$, also called
$A(1/0)$ in the Cartan-Kac classification. The Lie sub-algebra
$\mathcal{A}_{0}$ is $SU(2)U(1)$, and the odd $\mathcal{A}_{0}$ module
$\mathcal{A}_{1}$ is a complex doublet with $U(1)$ charge $-1$. The
fundamental representation [SNR77] is of superdimension (2/1) and fits the
leptons [N79, F79]. In the $(\nu_{L},\;e_{L};\;e_{R})$ basis, the 4 even
generators read:

$\lambda_{1}=\pmatrix{0&1&0\cr 1&0&0\cr
0&0&0}\,,\;\;\lambda_{2}=\pmatrix{0&-i&0\cr i&0&0\cr
0&0&0}\,,\;\;\lambda_{3}=\pmatrix{1&0&0\cr 0&-1&0\cr
0&0&0}\,,\;\;\lambda_{8}=\pmatrix{1&0&0\cr 0&1&0\cr 0&0&2}\,.$ (1)

the 4 odd generators read:

$\lambda_{4}=\pmatrix{0&0&1\cr 0&0&0\cr
1&0&0}\,,\;\;\lambda_{5}=\pmatrix{0&0&-i\cr 0&0&0\cr
i&0&0}\,,\;\;\lambda_{6}=\pmatrix{0&0&0\cr 0&0&1\cr
0&1&0}\,,\;\;\lambda_{7}=\pmatrix{0&0&0\cr 0&0&-i\cr 0&i&0}\,,$ (2)

and the chirality operator is

$\begin{array}[]{c}\chi=diag(1,1,-1)\,.\end{array}$ (3)

By inspection, we find that the anticommutator of the odd matrices close on
the even matrices, defining the symmetric structure constants $d^{a}_{ij}$ and
that the commutators of the odd matrices close on $\chi$ times the even
matrices, defining the skew-symmetric structure constants $f^{a}_{ij}$:

$\begin{array}[]{c}\lambda_{i}\,\lambda_{j}+\lambda_{j}\,\lambda_{i}=d^{a}_{ij}\,\lambda_{a}\;,\\\
\lambda_{i}\,\lambda_{j}-\lambda_{j}\,\lambda_{i}=i\;\chi\;f^{a}_{ij}\,\lambda_{a}\;,\\\
a=1,2,3,8\,\,;\;\;\;i,j=4,5,6,7\end{array}$ (4)

in these conventions, the $f^{a}_{ij}$ and $d^{a}_{ij}$ constants are real.
Finally, the vacuum $v$ is chosen along the $\lambda_{6}$ direction, whose
centralizer is the photon:

$\hbox{photon}=-(\lambda_{6})^{2}={1\over
2}\;(\lambda_{3}-\lambda_{8})=\pmatrix{0&0&0\cr 0&-1&0\cr 0&0&-1}\,.$ (5)

There is no loss of generality in this choice of $v$, it just corresponds to
the way we named the particles. If we rotate $v$ we just have to rotate the
name of the electron and the neutrino and keep the photon in the direction of
$v^{2}$. Notice that, with respect to the $SU(2/1)$ supermetric :

$\begin{array}[]{c}Str(.)=Tr(\chi\;.)\\\
\widehat{g}_{MN}=1/2\;STr(\lambda_{M}\lambda_{N})\;,\;\;M,N=1,2...8\\\
\widehat{g}_{ab}=diag(1,1,1,-1)\,,\qquad\widehat{g}_{45}=-\widehat{g}_{54}=\widehat{g}_{67}=-\widehat{g}_{76}=i\end{array}$
(6)

the photon is on the light-cone of the superalgebra.

The even generator of antilepton representation, in the
$({\overline{(e_{R})}}_{L};{\overline{(e_{L})}}_{R},{\overline{(\nu_{L})}}_{R})$
basis, are:

$\lambda_{1}=\pmatrix{0&0&0\cr 0&0&1\cr
0&1&0}\,,\;\;\lambda_{2}=\pmatrix{0&0&0\cr 0&0&-i\cr
0&i&0}\,,\;\;\lambda_{3}=\pmatrix{0&0&0\cr 0&1&0\cr
0&0&-1}\,,\;\;\lambda_{8}=\pmatrix{-2&0&0\cr 0&-1&0\cr 0&0&-1}\,.$ (7)

the 4 odd generators:

$\lambda_{4}=\pmatrix{0&0&-1\cr 0&0&0\cr
1&0&0}\,,\;\;\lambda_{5}=\pmatrix{0&0&i\cr 0&0&0\cr
i&0&0}\,,\;\;\lambda_{6}=\pmatrix{0&1&0\cr-1&0&0\cr
0&0&0}\,,\;\;\lambda_{7}=\pmatrix{0&-i&0\cr-i&0&0\cr 0&0&0}\,,$ (8)

and the chirality operator is

$\begin{array}[]{c}\chi=diag(1,-1,-1)\,.\end{array}$ (9)

Equations (3.4-5) are again true in this representation. Notice however that
the odd matrices (3.8) are anti-Hermitian, this is necessary to maintain (3.5)
since the electric charge of the antielectron is positive. Notice also that
the charge and parity (CP) are linked. The doublet and antidoublet must be of
opposite parity with respect to $\chi$ (3.3) and (3.9) in order to maintain
the commutator (3.4b). This switches the sign of the supermetric (3.6), as
befits a CP invariant theory.

##  4 The $SU(2/1)$ quark representation

In the standard model [Wei67, Sa68, GIM70, BIM72], each family of quarks
consists of 4 states, a left doublet and two right singlets, i.e.
$(u_{R}/(u_{L},d_{L})/d_{R})$ for the ’electron family’. Amazingly, the second
smallest irreducible representation of the $SU(2/1)$ superalgebra corresponds
to this chiral decomposition, it contains a free parameter $n$ which allows us
to fix the charge of the up quark. From the mathematical point of view
[SNR77], the existence of this 4 dimensional representation is a simple
consequence of the isomorphism between $SU(2/1)$ and $OSp(2/2)$, which is a
generalization of the well known isomorphisms between the first members of the
infinite families of simple Lie algebras. But from the physical point of view,
it came as a great surprise. In the original papers [F79, N79], the quarks
were respectively left out and listed as a counterargument. The incorporation
a few weeks later of the quarks [DJ79, NT80] really appeared as a significant
confirmation of the model. The 4 even generators read:

$\lambda_{1}=\pmatrix{0&0&0&0\cr 0&0&1&0\cr 0&1&0&0\cr
0&0&0&0}\,,\;\;\lambda_{2}=\pmatrix{0&0&0&0\cr 0&0&-i&0\cr 0&i&0&0\cr
0&0&0&0}\,,\;\;$ (10) $\lambda_{3}=\pmatrix{0&0&0&0\cr 0&1&0&0\cr 0&0&-1&0\cr
0&0&0&0}\,,\;\;\lambda_{8}={1\over n}\;\pmatrix{-n-1&0&0&0\cr 0&-1&0&0\cr
0&0&-1&0\cr 0&0&0&n-1}\,.$ (11)

The 4 odd generators are not Hermitian, for any value of $n$, they read:

$\begin{array}[]{c}\lambda_{4}={1\over\sqrt{2n}}\pmatrix{0&0&-\sqrt{n+1}&0\cr
0&0&0&\sqrt{n-1}\cr\sqrt{n+1}&0&0&0\cr 0&\sqrt{n-1}&0&0}\,,\\\ \\\
\lambda_{5}={1\over\sqrt{2n}}\pmatrix{0&0&i\sqrt{n+1}&0\cr
0&0&0&-i\sqrt{n-1}\cr i\sqrt{n+1}&0&0&0\cr 0&i\sqrt{n-1}&0&0}\,,\\\ \\\
\lambda_{6}={1\over\sqrt{2n}}\pmatrix{0&\sqrt{n+1}&0&0\cr-\sqrt{n+1}&0&0&0\cr
0&0&0&\sqrt{n-1}\cr 0&0&\sqrt{n-1}&0}\,,\\\ \\\
\lambda_{7}={1\over\sqrt{2n}}\pmatrix{0&-i\sqrt{n+1}&0&0\cr-i\sqrt{n+1}&0&0&0\cr
0&0&0&-i\sqrt{n-1}\cr 0&0&i\sqrt{n-1}&0}\,.\end{array}$ (12)

Finally the chirality operator is

$\begin{array}[]{c}\chi=diag(-1,1,1,-1)\,.\end{array}$ (13)

We have normalized the quark matrices (4.1-4.4) so that they have exactly the
same supercommutators as the lepton matrices (3.1,3.2). As desired, the super-
Killing metric (3.6) is also identical in the lepton and the quark
representation. Notice however that the commutators of the odd quark matrices
do not close on $\chi$ times the even matrices.

Since $n$ is a free parameter, $SU(2/1)$ does not predict the charge of the
quarks, or the number of colors. However, the superalgebra does relate the
$U(1)$ charges of the singlets and a number of desired relations (post-
dictions) are automatically satisfied.

First, for $n=-1$ (or $n=1$), we recover the lepton (or antilepton)
representation with only 3 states. In other words, $SU(2/1)$ implies that if
the charge of the electron is equal to the charge of the $W$ vector, the
$U(1)$ neutral right-neutrino is also neutral with respect to the odd
generators.

Then, we should impose the cancellation of the triangle anomalies.

$\begin{array}[]{c}d_{abc}=STr_{f}(\lambda_{a}(\lambda_{b}\lambda_{c}+\lambda_{c}\lambda_{b}))=0\,,\;\;\;a,b,c=1,2,3,8\;,\end{array}$
(14)

If we consider a lepton family, with one lepton of $U(1)$ charge $1$, and
$n^{\prime}$ quarks of $U(1)$ charge $-1/n$, the $SU(2)^{2}U(1)$ anomaly
vanishes if and only if we choose $n=n^{\prime}$. But although there are no
other adjustable parameters, $SU(2/1)$ then implies that the $U(1)^{3}$
anomaly also vanishes

$\begin{array}[]{c}STr_{f}\;(\lambda_{8}^{3})=2-8+n\;((n+1)^{3}-2-(n-1)^{3})/n^{3}=0\end{array}$
(15)

for any value of $n$. The vacuum and the photon direction were already chosen
in the lepton representation. In the same conventions (3.5) where the electric
charge of the electron is -1, we find:

$\hbox{photon}=-(\lambda_{6})^{2}={1\over
2}\;(\lambda_{3}-\lambda_{8})={1\over 2n}\;\pmatrix{n+1&0&0&0\cr 0&n+1&0&0\cr
0&0&1-n&0\cr 0&0&0&1-n}\,.$ (16)

Choosing the correct number of colors $n=3$ implies the correct electric
charges for the $up$ quark $2/3$ and the $down$ quark $-1/3$ . Finally, the
odd sector antisymmetric structure constants and trace metric also vanish:

$\begin{array}[]{c}f_{aij}=Tr_{f}(\lambda_{a}(\lambda_{i}\lambda_{j}-\lambda_{j}\lambda_{i}))=0\,,\;\;\;a=1,2,3,8\,,\\\
g_{ij}={1\over
2}Tr_{f}(\lambda_{i}\lambda_{j})=0\,,\;\;\;i,j=4,5,6,7\,.\end{array}$ (17)

We will now study in the next sections some consequences of these identities.

##  5 The scalar propagator

We would like to construct a quantum field theory, based on the $SU(2/1)$
superalgebra, which would extend the Yang-Mills theory [YM54] associated to
its maximal $SU(2)U(1)$ Lie sub-algebra and incorporate in some way the odd
matrices. In [TM06], we have shown that we can construct an associative
covariant exterior differential, mixing left and right chiral Fermions, if and
only if the Fermion form a representation of a Lie superalgebra, graded by the
chirality. As we have just seen, these conditions are met exactly by the
existing fundamental Fermions: the leptons and the quarks. The connection is
then defined as the sum of the usual Yang-Mills 1-form $A_{\mu}\;dx^{\mu}$
plus the constant Dirac 1-form $\gamma_{\mu}\;dx^{\mu}$ multiplied by scalar
field valued in the odd sector of the superalgebra.

The natural idea would therefore be to introduce a scalar-field $\Phi$ with a
minimal coupling to the Fermions of the form
$\Psi^{\dagger}\;\Phi^{i}\lambda_{i}\;\Psi$. However, this is not directly
possible since the $\lambda_{i}$ are not Hermitian. To fix this problem, we
postulate that the field $\Phi^{i}$ is composed of 2 parts, each with
intrinsically chiral couplings. Considering the chirality operator $\chi$
(3.3,4.4) with eigenvalue $1$ $(-1)$ on the left (right) Fermions, we define
the left and right projectors $p_{L}=(1+\chi)/2$ and $p_{R}=(1-\chi)/2$, and
define, for any matrix M, the chiral traces.

$\begin{array}[]{c}Tr_{L}(M)=Tr(p_{L}\;M)\;,\;\;Tr_{R}(M)=Tr(p_{R}\;M)\;,\;\;Str(M)=Tr(\chi\;M)\;.\end{array}$
(18)

We then decompose the odd $\lambda$ matrices as

$\begin{array}[]{c}\lambda_{i}=\lambda_{iL}+\lambda_{iR}\\\
\lambda_{iL}=p_{L}\;\lambda_{i}=\lambda_{i}\;p_{R}\;,\;\;\lambda_{iR}=p_{R}\;\lambda_{i}=\lambda_{i}\;p_{L}\;\;,\end{array}$
(19)

and assume the existence of 2 scalar fields $\Phi_{L}$ which absorbs right
Fermions and emits left ones and $\Phi_{R}$ which absorbs left Fermions and
emits right ones, according to the coupling

$\begin{array}[]{c}\Psi^{\dagger}_{L}\;\Phi^{i}_{L}\lambda_{iL}\;\Psi_{R}\;+\Psi^{\dagger}_{R}\;\Phi^{i}_{R}\lambda_{iR}\;\Psi_{L}\;.\end{array}$
(20)

To find the correct propagator of the $\Phi$ field, we compute the quantum
corrections due to the insertion of a Fermion loop between 2 $\Phi$ states.

This induces a scalar counterterm of the form

$\begin{array}[]{c}\mathcal{L}_{s}=\eta_{ij}\;\partial_{\mu}\Phi^{i}_{L}\;\partial_{\mu}\Phi^{j}_{R}\end{array}$
(21)

where

$\begin{array}[]{c}\eta_{ij}={1\over
n+1}Tr_{L}(\lambda_{i}\;\lambda_{j})={1\over
n+1}Tr_{R}(\lambda_{j}\;\lambda_{i})\,.\end{array}$ (22)

There are two contributions, one coming from the lepton loop and one from the
quark loop

$\begin{array}[]{c}(\eta_{ij})_{lepton}={1\over n+1}\delta_{ij}+{i\over
n+1}\;\epsilon_{ij}\;,\\\ (\eta_{ij})_{quark}=-{1\over
n(n+1)}\delta_{ij}+{i\over n+1}\;\epsilon_{ij}\,,\end{array}$ (23)

where $\epsilon_{45}=-\epsilon_{54}=\epsilon_{67}=-\epsilon_{76}=1$, all other
components being zero, which coincides with the odd part of (3.6). Summing
over a whole family $f$ with $n$ colored quarks, we find that the Fermion loop
is proportional to the canonical supermetric of $SU(2/1)$:

$\begin{array}[]{c}(\eta_{ij})_{f}=i\;\epsilon_{ij}\;=\widehat{g}_{ij}\end{array}$
(24)

For the moment, we have a weird scalar model, with an antisymmetric propagator
and non Hermitian couplings. But now a miracle occurs. If we change variables
as follows, the Lagrangian becomes canonical. Consider the two $SU(2/1)$ odd
valued scalar fields $H$ and $K$:

$\begin{array}[]{c}\Phi_{iL}=a_{i}+i\epsilon_{ij}b_{j}\;,\;\;a_{i}={1\over
2}\;(H_{i}+i\;K_{i})\;,\\\ \\\
\Phi_{iR}=a_{i}-i\epsilon_{ij}b_{j}\;,\;\;b_{i}={1\over
2}\;(H_{i}-i\;K_{i})\;,\end{array}$ (25)

where $a_{i}$ and $b_{i}$ are convenient intermediate variables which are not
reused below. Substituting (5.8) in (5.4) we obtain the canonical propagator

$\begin{array}[]{c}\mathcal{L}_{s}=i\;\epsilon_{ij}\;\partial_{\mu}\Phi^{i}_{L}\;\partial_{\mu}\Phi^{j}_{R}\;={1\over
2}\delta_{ij}\;(\partial_{\mu}H^{i}\;\partial_{\mu}H^{j}\;+\partial_{\mu}K^{i}\;\partial_{\mu}K^{j}\;)\,.\end{array}$
(26)

We then define the $\mu^{\pm}$ Hermitian matrices:

$\begin{array}[]{c}\lambda_{i}=(\mu^{-}_{i}-i\;\mu^{+}_{i})\,,\qquad\mu^{-}_{i}={1\over
2}\;(\lambda_{i}+\lambda_{i}^{\dagger})\,,\\\
\lambda_{i}^{\dagger}=(\mu^{-}_{i}+i\;\mu^{+}_{i})\,,\qquad\mu^{+}_{i}={i\over
2}\;(\lambda_{i}-\lambda_{i}^{\dagger})\,,\end{array}$ (27)

and rewrite the scalar-Fermion interaction in terms of these new variables:

$\Psi^{\dagger}\;(\Phi^{i}_{L}\;\lambda_{iL}+\Phi^{i}_{R}\;\lambda_{iR})\;\Psi=\Psi^{\dagger}\;(H^{i}\;\mu^{-}_{i}+K^{i}\;\mu^{+}_{i})\;\Psi\,.$
(28)

The matrices $\mu^{\pm}_{i}$ split according to the electric charge and are
Hermitian. The field $H^{i}$ interacts only with the negatively charged right
singlets (electron, down quark) and the field $K^{i}$ just with the positively
charged right singlets (up quark). In the lepton representation, the
$\mu^{-}_{i}$ matrices are equal to the Hermitian matrices given in (3.2)

$\begin{array}[]{c}\mu^{-}_{4}=\pmatrix{0&0&1\cr 0&0&0\cr
1&0&0}\,,\;\;\mu^{-}_{5}=\pmatrix{0&0&-i\cr 0&0&0\cr i&0&0}\,,\\\ \\\
\mu^{-}_{6}=\pmatrix{0&0&0\cr 0&0&1\cr
0&1&0}\,,\;\;\mu^{-}_{7}=\pmatrix{0&0&0\cr 0&0&-i\cr 0&i&0}\,.\end{array}$
(29)

In the quark representation, the $\mu^{-}_{i}$ are equal to the right and low
Hermitian corner of the matrices (4.3)

$\begin{array}[]{c}\mu^{-}_{4}=\sqrt{n-1\over 2n}\;\pmatrix{0&0&0&0\cr
0&0&0&1\cr 0&0&0&0\cr 0&1&0&0}\,,\;\;\mu^{-}_{5}=\sqrt{n-1\over
2n}\;\pmatrix{0&0&0&0\cr 0&0&0&-i\cr 0&0&0&0\cr 0&i&0&0}\,,\\\ \\\
\mu^{-}_{6}=\sqrt{n-1\over 2n}\;\pmatrix{0&0&0&0\cr 0&0&0&0\cr 0&0&0&1\cr
0&0&1&0}\,,\;\;\mu^{-}_{7}=\sqrt{n-1\over 2n}\;\pmatrix{0&0&0&0\cr 0&0&0&0\cr
0&0&0&-i\cr 0&0&i&0}\,.\end{array}$ (30)

The top left corner of the $\mu^{+}_{i}$ matrices are proportional to the
antielectron matrices (3.8)

$\begin{array}[]{c}\mu^{+}_{4}={\sqrt{n+1\over 2n}}\;\pmatrix{0&0&i&0\cr
0&0&0&0\cr-i&0&0&0\cr 0&0&0&0}\,,\;\;\mu^{+}_{5}={\sqrt{n+1\over
2n}}\;\pmatrix{0&0&1&0\cr 0&0&0&0\cr 1&0&0&0\cr 0&0&0&0}\,,\\\ \\\
\mu^{+}_{6}={\sqrt{n+1\over 2n}}\;\pmatrix{0&-i&0&0\cr i&0&0&0\cr 0&0&0&0\cr
0&0&0&0}\,,\;\;\mu^{+}_{7}={\sqrt{n+1\over
2n}}\;\pmatrix{0&-1&0&0\cr-1&0&0&0\cr 0&0&0&0\cr 0&0&0&0}\,.\end{array}$ (31)

If we sum over a complete family, we find that the matrices associated to $H$
and $K$ have the same trace-metric:

$Tr(\mu^{+}_{i}\mu^{-}_{j})=0\;,\;\;Tr(\mu^{+}_{i}\mu^{+}_{j})=Tr(\mu^{-}_{i}\mu^{-}_{j})=(n+1)\;\delta_{ij}\,,\qquad
i,j=4,5,6,7\,.$ (32)

This identity, like (4.8), is a consequence of the vanishing of the sum of the
charges of all the right singlets (electron + n up quarks + n down quarks) of
a family. It shows that one can compute the renormalization of the scalar
propagator either using the $\Phi_{L},\;\Phi_{R}$ fields (5.4) or using the
$H,\;K$ fields (5.8), and obtain the same result (5.9).

##  6 The scalar potential

From now on, we restrict our attention to the case $n=3$ colors. Consider the
structure of the scalar potential induced by a Fermion loop. The cubic terms
vanish, since each interaction with a scalar changes the chirality of the
Fermions. The term of degree 6 or higher converge. So we only have to compute
the quartic potential. If we work in terms of the $\Phi$ fields, the
counterterm is proportional to

$\begin{array}[]{c}\Phi^{i}_{L}\;\Phi^{j}_{R}\;\Phi^{k}_{L}\;\Phi^{l}_{R}\;\;Tr_{L}(\lambda_{i}\;\lambda_{j}\;\lambda_{k}\;\lambda_{l}\;)\;.\end{array}$
(33)

Generalizing (5.6), we find that many unfriendly terms, for example
$\Phi^{4}_{L}\;\Phi^{4}_{R}\;\Phi^{4}_{L}\;\Phi^{5}_{R}$, are eliminated by
the balance between the leptons and the quarks. If we introduce the metric:

$\widetilde{G}_{ab}=\pmatrix{2\delta_{ab}-\alpha\;\widehat{g}_{ab}&-\widehat{g}_{ab}\cr-\widehat{g}_{ab}&2\delta_{ab}-\alpha\;\widehat{g}_{ab}}\;,$
(34)

and the products

$\begin{array}[]{c}Z^{a}_{L}=(f^{a}_{ij}-i\;d^{a}_{ij})\;\Phi^{i}_{L}\,\Phi^{j}_{R}\;,\\\
Z^{a}_{R}=(f^{a}_{ij}+i\;d^{a}_{ij})\;\Phi^{i}_{L}\,\Phi^{j}_{R}\;.\end{array}$
(35)

where the $d^{a}_{ij}$ and the $f^{a}_{ij}$ are defined in (3.4), the direct
calculation of the Fermion loop induces, up to an infinite multiplicative
constant, a scalar potential of the form

$\begin{array}[]{c}V={1\over
8}\;\pmatrix{Z^{a}_{L}&Z^{a}_{R}}\;\widetilde{G}_{ab}\;\pmatrix{Z^{a}_{L}\cr
Z^{a}_{R}}\;.\end{array}$ (36)

The $\alpha$ parameter is arbitrary in (6.4), because we have the identity

$Z^{a}_{L}\;\widehat{g}_{ab}\;Z^{b}_{L}=0\;.$ (37)

If we work with the $\mu^{\pm}$ Hermitian matrices (5.12-14), we find,
generalizing (5.15), that the symmetrized quartic traces are identical

$\begin{array}[]{c}Tr((\mu^{+}_{i}\mu^{+}_{j}+\mu^{+}_{j}\mu^{+}_{i})(\mu^{+}_{k}\mu^{+}_{l}+\mu^{+}_{l}\mu^{+}_{k}))\\\
=Tr((\mu^{-}_{i}\mu^{-}_{j}+\mu^{-}_{j}\mu^{-}_{i})(\mu^{-}_{k}\mu^{-}_{l}+\mu^{-}_{l}\mu^{-}_{k}))\\\
=32/9\;(\delta_{ij}\delta_{kl}+\delta_{ik}\delta_{jl}+\delta_{il}\delta_{jk})\;.\end{array}$
(38)

Finally the mixed traces satisfy, for all $i,j,k,l$:

$\begin{array}[]{c}Tr(\mu^{+}_{i}\mu^{-}_{j})=Tr(\mu^{+}_{i}\mu^{-}_{j}\mu^{-}_{k}\mu^{-}_{l})=Tr(\mu^{+}_{i}\mu^{+}_{j}\mu^{+}_{k}\mu^{-}_{l})=Tr(\mu^{+}_{i}\mu^{-}_{j}\mu^{+}_{k}\mu^{-}_{l})=0\,,\\\
\\\ Tr(\mu^{+}_{i}\mu^{+}_{i}\mu^{-}_{i}\mu^{-}_{i})=0\;,\qquad\hbox{for any
i, no sum implied.}\end{array}$ (39)

If we now introduce the symmetric product

$\begin{array}[]{c}(X*X)^{a}=d^{a}_{ij}\;X^{i}\,X^{j}\;;\;\;\;X=H,\;K\;,\end{array}$
(40)

the direct calculation of the Fermion loop induces, up to the same infinite
multiplicative constant, a scalar potential of the form

$V={1\over
16}\;\pmatrix{(H*H)^{a}&(K*K)^{a}}\pmatrix{\widetilde{G}_{ab}}\pmatrix{(H*H)^{b}\cr(K*K)^{b}}\;.$
(41)

The $\alpha$ parameter (6.2) is arbitrary in (6.9), because in any Lie
superalgebra, we have the identity

$\widehat{g}_{ab}(X*X)^{a}\;(X*X)^{b}=0$ (42)

Notice that for the same reason, the mixed $H\;K$ term vanishes if and only if
$H$ and $K$ are parallel. This identity insures that the centralizer of the
true vacuum is nontrivial, or in other words, that the photon remains
massless.

If we change variable again and define

$\begin{array}[]{c}u_{1}=(H_{4}+iH_{5})/\sqrt{2}\;,\\\
u_{2}=(H_{6}+iH_{7})/\sqrt{2}\;,\\\ v_{1}=(K_{4}+iK_{5})/\sqrt{2}\;,\\\
v_{2}=(K_{6}+iK_{7})/\sqrt{2}\;,\end{array}$ (43)

and substitute in (5.9) and (6.9), we recover up to a scale factor $\sigma$
the propagator and potential of Fayet [F74, F75]

$\begin{array}[]{c}\mathcal{L}_{s}+V=\partial_{\mu}\overline{u}\;\partial_{\mu}u+\partial_{\mu}\overline{v}\;\partial_{\mu}v\;+\\\
+{\sigma}((\overline{u}u)^{2}+(\overline{v}v)^{2}+(\overline{u}_{1}\overline{v}_{2}-\overline{u}_{2}\overline{v}_{1})(u_{1}v_{2}-u_{2}v_{1}))\;.\end{array}$
(44)

In these variable, it is immediately clear that, assuming a symmetry breaking
negative mass term for the scalars, the potential is minimal when $u$ and $v$
are parallel, and therefore the photon remains massless. The scalar potential
(6.9) is invariant under the infinitesimal transformation

$\begin{array}[]{c}\Delta\;H_{i}=(H^{k}\;\delta_{kl}\;K^{l})\;\widehat{g}_{ij}\;K^{j}-(H^{k}\;\widehat{g}_{kl}\;K^{l})\;\delta_{ij}\;K^{j}\;\;,\\\
\Delta\;K_{i}=(H^{k}\;\delta_{kl}\;K^{l})\;\widehat{g}_{ij}\;H^{j}-(H^{k}\;\widehat{g}_{kl}\;K^{l})\;\delta_{ij}\;H^{j}\;\;,\end{array}$
(45)

or, if we use the variables $u$ and $v$:

$\begin{array}[]{c}\Delta\;u=(u.{\overline{v}})\;v\;\;,\;\;\;\Delta\;v=-(v.{\overline{u}})\;u\;\;,\\\
\Delta\;{\overline{v}}=(u.{\overline{v}})\;{\overline{u}}\;\;,\;\;\;\Delta\;{\overline{u}}=-(v.{\overline{u}})\;{\overline{v}}\;\;,\end{array}$
(46)

whereas the term $(u.v)({\overline{u}}.{\overline{v}})$ which could give a
mass to the photon and is absent from our Lagrangian, is not invariant under
$\Delta$.

##  7 Scalar Ward identities

The study of the counterterms induced by the Fermion loop has given in the
previous section the structure of the propagator and of the potential, but not
their scale. If we write our Lagrangian as

$\begin{array}[]{c}\mathcal{L}=-{1\over 4}(F^{a}_{\mu\nu})^{2}-{1\over
2}((D_{\mu}H)^{2}+(D_{\mu}K)^{2})+\lambda\;{g^{2}}\;V+{\overline{\Psi}}\;{g\over
2}\;(\gamma^{\mu}\;A_{\mu}+\xi\Phi\;)\;\Psi\;,\end{array}$ (47)

The factor $2$ in $g/2$ is needed because our $SU(2)$ matrices (3.1) are
normalized as usual in mathematics, not as the standard spin $1/2$ isospin
matrices of physics. The weak angle and the coefficients $\lambda$ and $\xi$
are not yet known. We will now fix them by requiring that they do not depend
on the number of lepton families.

The only contribution of the other families to the coupling of the vectors and
scalars to the Fermion of the first family is through the renormalization of
the wave function of the vector and the scalar. The counterterms are
proportional to

$\begin{array}[]{c}Z(A_{a}^{2})={2\over 3}\;{g^{2}\over
4}\;Tr_{f}\;\lambda_{a}^{2}\;\;,\;\;\;Z(A_{8}^{2})=tg^{2}\theta\;{2\over
3}{g^{2}\over
4}\;Tr_{f}\;\lambda_{a}^{2}\;\;,\;\;\;Z(H^{2})=\xi^{2}\;{g^{2}\over
4}\;Tr_{f}\;\mu_{i}^{2}\;\;.\end{array}$ (48)

The factor $2/3$ in the vectors comes from the kinematic contribution of the
Dirac matrices. We count $2/3$, not the usual $4/3$ of a massive Fermion,
since here the left and right spinors contribute separately to the trace term.
Over a full family, the $SU(2)$ even matrices (3.1,4.1) are normalized to
$Tr_{f}(\lambda_{a}\;\lambda_{b})=8\delta_{ab}\;$. The $U(1)$ matrix is
normalized to $Tr_{f}(\lambda_{8}\;\lambda_{8})=40/3$, and the odd matrices to
$Tr_{f}(\mu^{\pm}_{i}\;\mu^{\pm}_{j})=4\;\delta_{ij}$. To integrate these 3
counterterms in the renormalization of the coupling constant $g$, and insure
that $tg\theta$ and $\xi$ are not renormalized, we fix

$\begin{array}[]{c}tg^{2}\theta={3\over 5}\;\;,\;\;\;sin^{2}\theta={3\over
8}\;\;,\;\;\;\xi={2\over\sqrt{3}}\;\;.\end{array}$ (49)

The weak angle coincides with the value derived from the $SU(5)$ grand unified
theory [GG74]. We then look at the renormalization of the scalar potential.
The standard Lie algebra Ward identities ensure that the contribution of the
Fermions to the self coupling of the vectors, the $g^{2}\;A^{4}$ term, can be
reabsorbed in the renormalization of $g^{2}$. A similar condition holds for
the scalar potential $\lambda\;g^{2}\;H^{4}$.

$\begin{array}[]{c}Z(V)={1\over 3}\,{(g\xi)^{4}\over\lambda g^{2}}\end{array}$
(50)

The contribution of the Fermions is reabsorbed in the renormalization of
$g^{2}$ and therefore $\lambda$ is not renormalized if we choose

$\begin{array}[]{c}Z(V)=Z(H^{2})==>\lambda={\xi^{2}\over 3}={4\over
9}\end{array}$ (51)

However, we note that the Bosonic counterterms, induced by the vector and
scalar 1-loops perturb all the scalar couplings, except the very structure of
the scalar potential which guarantees the survival of an unbroken $U(1)$ and
the existence of the massless photon. Furthermore the strong interactions
gluon vector fields interact with the quarks, but not with the leptons. The
photon-quark coupling is protected by a Ward identity and is not renormalized
by the gluon-quark interaction, but the scalar-quark vertex is not protected.
Hence, the balance between quarks and leptons, which is central to our
analysis is broken by these counterterms. This problem can either be
interpreted as a fatal flaw of the $SU(2/1)$ model or as sign that the quantum
theory is still incomplete.

##  8 The mass of the Higgs

Since we have now fixed the scale of the potential we can derive the mass of
the physical scalars, the two neutral fields $H^{0}_{1or2}$ and the charged
field $H^{\pm}$. Suppressing the Fermions, the Lagrangian reads:

$\begin{array}[]{c}\mathcal{L}=-{1\over 4}(F^{a}_{\mu\nu})^{2}-{1\over
2}((D_{\mu}H)^{2}+(D_{\mu}K)^{2})+{\lambda}\;V\end{array}$ (52)

where $V$ is given in (6.9). If we develop the potential and show explicitly
the occurrences of the component fields $H_{4}$. We find:

$\begin{array}[]{c}{\lambda}\;V={\lambda\over
4}\;(H^{4}_{4}+H^{2}_{4}\;(2\;(H^{2}_{5}+H^{2}_{6}+H^{2}_{7})+K^{2}_{6}+K^{2}_{7})+\\\
+2\;H_{4}H_{6}\;(K_{4}K_{6}+K_{5}K_{7})+2\;H_{4}H_{7}\;(K_{4}K_{7}+K_{5}K_{6}))+...\end{array}$
(53)

Three points should be noticed. The $\lambda$ parameter is scaled as usual to
give $\lambda/4\;H^{4}_{4}$. There are no terms $H^{2}_{4}\;K^{2}_{4}$ or
$H^{2}_{4}\;K^{2}_{5}$, finally the coefficients of $H_{4}$ are invariant
under a $\lambda_{8}$ rotation of the $K$ fields, which rotates $K_{4}$ into
$K_{5}$ and $K_{6}$ into $K_{7}$.

We now add by hand in the Lagrangian a quadratic symmetry breaking ’imaginary
mass’ term

$\begin{array}[]{c}-{\lambda\over
2}\;v^{2}\;(sin^{2}\beta\;H^{2}+cos^{2}\beta\;K^{2})\;.\end{array}$ (54)

We assume that the vacuum expectation value of $H$ is in the direction
$H_{6}$. As discussed in section 3, this choice just fixes the name of the
particles. To minimize the mixed $H^{2}\;K^{2}$ potential, the $K$ vacuum
expectation value must be in the plane $(K_{6},K_{7})$. The vacuum correctly
preserves the photon (3.5, 4.7). We then rotate $<K>$ to $<K_{6}>$. This
’phase’ invariance corresponds to the fact that one of the 4 neutral states in
the hyper-plane $(H_{6},H_{7},K_{6},K_{7})$, usually called the ’$A_{0}$’
pseudo-scalar, is absent from the quartic potential and its mass can be fixed
arbitrarily in the symmetry-breaking term. We minimize the potential and find:

$\begin{array}[]{c}<H_{6}>=v\;sin\beta\;\;,\;\;\;<K_{6}>=v\;cos\beta\;\;,\;\;\;M_{W}=v\;g/2\;,\end{array}$
(55)

where $\beta$ is defined modulo $\pi/2$. We then expand the potential around
the vacuum to second order in $v$. We find 4 massless states which produce the
$A_{0}$ and the longitudinal components of the massive vector fields $Z$ and
$W^{\pm}$

$\begin{array}[]{c}H_{7}\;,\;\;K_{7}\;,\;\;(sin\beta\;H_{4}+cos\beta K_{4})\pm
i(sin\beta\;H_{5}+cos\beta K_{5})\;,\end{array}$ (56)

two charged physical scalars

$\begin{array}[]{c}H^{\pm}=(cos\beta\;H_{4}-sin\beta K_{4})\pm
i(cos\beta\;H_{5}-sin\beta K_{5})\;,\\\
M_{H^{\pm}}={\sqrt{2\;\lambda}}\;M_{W}=75.8\;GeV\end{array}$ (57)

and two neutral scalars in the $(H_{6},K_{6})$ plane, usually called
$H^{0}_{1}$ and $H^{0}_{2}$, with masses

$\begin{array}[]{c}M_{H^{0}_{1}}={2\;\sqrt{2\;\lambda}}\;cos\beta\;M_{W}=({\sqrt{2}}cos\beta)\;107.2\;GeV\;,\\\
M_{H^{0}_{2}}={2\;\sqrt{2\;\lambda}}\;sin\beta\;M_{W}=({\sqrt{2}}sin\beta)\;107.2\;GeV\;.\end{array}$
(58)

In the symmetric case $\beta=\pi/4$ the 2 neutral scalar are at $107.2\;GeV$.

##  9 The $SU(2/1)$ hyper-curvature

Since the scalar fields (5.3) play the role of a gauge field associated to the
odd generators of the superalgebra, it seems natural to introduce them as part
of the covariant differential and curvature 2-form. Such a construction was
proposed in [TM06], where we found that the covariant differential is
associative if and only if the Fermions sit in a representation of the
superalgebra graded by the chirality. As shown in sections 2 and 3, this
condition is met by the observed leptons and quarks. But we found in section 3
the additional complication that the quark representation is non Hermitian, so
the scalar field is complex, so effectively, the number of odd generators in
the quantum field theory is twice the number of odd generators of the
$SU(2/1)$ superalgebra.

We are going to show that the construction of the curvature can be generalized
and yields a structure we would like to call a H-curvature, by reference to
Fayet and Iliopoulos [F74, F75], who were the first to consider, in the
context of Hyper-Symmetry, a doubling of the Higgs field introduced by
Weinberg [Wei67] and to Sternberg and Wolf [SW78] who introduced the Hermitian
algebras.

Following [TM06], we define the Dirac 1-forms

$\begin{array}[]{c}\Gamma=dx^{\mu}\;\sigma_{\mu}\
+dx^{\mu}\;\overline{\sigma}_{\mu}\,.\end{array}$ (59)

where $\sigma$ maps the right Fermions on the left Fermions and vice versa. We
then define the chiral Yang-Mills 1-form $\widetilde{A}$ and covariant
differential $\widetilde{D}$

$\begin{array}[]{c}\widetilde{A}=A+\Phi_{L}+\Phi_{R}=dx^{\mu}\;A^{a}_{\mu}\;\lambda_{a}+\Gamma\;\Phi^{i}_{L}\lambda_{iL}+\Gamma\;\Phi^{i}_{R}\lambda_{iR}\qquad,\widetilde{D}=d+\widetilde{A}\;.\end{array}$
(60)

Notice that $\widetilde{A}$ is an exterior form of homogeneous degree 1, in
contradistinction with the Ne’eman Quillen connection [TMN82, Q85] sometimes
discussed in the $SU(2/1)$ literature ([NS90, NSF05] and references therein).
The curvature is defined as the square of the covariant differential

$\begin{array}[]{c}\widetilde{F}=\widetilde{D}\;\widetilde{D}=d\widetilde{A}+\widetilde{A}\;\widetilde{A}\;.\end{array}$
(61)

Since the $dx^{\mu}$ anticommute, the classical term

$\begin{array}[]{c}A\;A=dx^{\mu}dx^{\nu}\;A^{a}_{\mu}\;A^{b}_{\nu}\;\lambda_{a}\lambda_{b}\end{array}$
(62)

is antisymmetric in $ab$ and is internal provided the even matrices
$\lambda_{a}$ close by commutation

$\begin{array}[]{c}\lambda_{a}\lambda_{b}-\lambda_{b}\lambda_{a}=i\;f^{c}_{ab}\;\lambda_{c}\;.\end{array}$
(63)

Similarly, the scalar vector term in $\Gamma\;D\Phi$ generate the covariant
differential of the scalars provided the even odd sector close by commutation

$\begin{array}[]{c}\lambda_{a}\lambda_{i}-\lambda_{i}\lambda_{a}=i\;f^{j}_{ai}\;\lambda_{j}\;.\end{array}$
(64)

However, the structure of the scalar-scalar term is more involved. Since
$\Phi_{L}$ and $\Phi_{R}$ are independent, the $\lambda$ matrices are neither
symmetrized nor skew symmetrized and we get

$\begin{array}[]{c}{1+\chi\over
2}\;\Phi^{i}_{L}\Phi^{j}_{R}\;\lambda_{i}\lambda_{j}+{1-\chi\over
2}\;\Phi^{i}_{R}\Phi^{j}_{L}\;\lambda_{i}\lambda_{j}\;.\end{array}$ (65)

We now decompose the matrix product into the symmetric and skew symmetric
parts

$\begin{array}[]{c}\lambda_{i}\lambda_{j}={1\over
2}(\\{\lambda_{i},\lambda_{j}\\}+[\lambda_{i},\lambda_{j}])\end{array}$ (66)

and we find that the scalar-scalar term can be expressed as a linear
combination of even generators provided the anticommutator of the odd matrices
close on the even matrices and the commutators of the odd matrices close on
$\chi$ times the even matrices, defining the skew-symmetric structure
constants $f^{a}_{ij}$:

$\begin{array}[]{c}\lambda_{i}\,\lambda_{j}+\lambda_{j}\,\lambda_{i}=d^{a}_{ij}\,\lambda_{a}\;,\\\
\lambda_{i}\,\lambda_{j}-\lambda_{j}\,\lambda_{i}=i\;\chi\;f^{a}_{ij}\,\lambda_{a}\;.\end{array}$
(67)

Therefore the unsymmetrized product of the odd matrices satisfy

$\begin{array}[]{c}\lambda_{i}\,\lambda_{j}={1\over
2}(d^{a}_{ij}+i\;\chi\;f^{a}_{ij})\lambda_{a}\;.\end{array}$ (68)

By inspection (3.4), all these conditions are satisfied by the lepton
representation (3.1-3). In fact, the construction works with the fundamental
representation of any $SU(m/n)$ superalgebra, because all the generators
coincide with the generators of $SU(m+n)$ except for the supertraceless $U(1)$
but $\chi\;U(1)$ is traceless and coincides with the $U(1)$ of $SU(m+n)$.

The structure (9.9) was introduced by Sternberg and Wolf [SW78], who call it a
Hermitian algebra and discussed by Ne’eman and Sternberg in several subsequent
papers [NS90, NSF05], but its exact relevance to quantum field theory remained
unclear. The unusual presence of the chirality operator $\chi$ on the right
hand side of the commutator (9.9) was not explained, because they did not
consider the possibility of splitting the odd generators and doubling the
number of Higgs fields. But in our modified framework, if we compare (9.7) and
(9.10) and use $\chi^{2}=1$, we see that the $\chi$ operator present in (9.9)
disappears from the curvature 2-form which is now correctly valued on the even
matrices and can be written as

$\begin{array}[]{c}\widetilde{F}_{L}=\widehat{F}^{a}_{L}\;\lambda_{a}+\Gamma\;D\Phi^{i}_{L}\;\lambda_{iL}+\Gamma\;D\Phi^{i}_{R}\;\lambda_{iR}\;,\\\
\widehat{F}^{a}_{L}=dA^{a}+{i\over 2}f^{a}_{bc}\;A^{b}A^{c}+{i\over
2}\;\Gamma\Gamma\;\Phi^{i}_{L}\Phi^{j}_{R}\;(f^{a}_{ij}-i\;d^{a}_{ij})\;.\end{array}$
(69)

The Bianchi identity
$\widetilde{D}\;\widetilde{F}=(\widetilde{D}\widetilde{D})\;\widetilde{D}-\widetilde{D}\;(\widetilde{D}\widetilde{D})=0$
follows from the associativity of the matrix product and implies the Hermitian
Jacobi identity of [SW78]. We can also construct a second solution
$\widehat{F}_{R}$ by switching the sign of the $d$ constants changing the
chirality:

$\begin{array}[]{c}\widehat{F}^{a}_{R}=dA^{a}+{i\over
2}f^{a}_{bc}\;A^{b}A^{c}+{i\over
2}\;\Gamma\Gamma\;\Phi^{i}_{L}\Phi^{j}_{R}\;(f^{a}_{ij}+i\;d^{a}_{ij})\;.\end{array}$
(70)

We recover in (9.11-12) the object $Z_{L/R}$ introduced in (6.3)

$\begin{array}[]{c}\widehat{F}_{L/R}=F+{i\over
2}\;\Gamma\Gamma\;\;Z_{L/R}\end{array}$ (71)

##  10 The electro-weak angle

From the definition of the classical curvature 2-form given in the previous
section, we can construct the Lagrangian in the usual way, using the Hodge
adjunction ∗ and the quadratic form $\widetilde{G}_{ab}$ (6.2) which
reproduces the scalar potential induced by the Fermion loops. The Lagrangian
can be written as:

${\mathcal{L}}_{F}={1\over
8\;g^{2}}Tr\;\pmatrix{{}^{*}\widehat{F}^{a}_{L}&{}^{*}\widehat{F}^{a}_{R}}\widetilde{G}_{ab}\pmatrix{\widehat{F}^{b}_{L}\cr\widehat{F}^{b}_{R}}$
(72)

Since the new bilinear scalar term in the H-curvature (9.13) coincides with
(6.3), the scalar potential is proportional to (6.4) which can be rewritten as
(6.9). It is curious to see that the combined effect of the leptons and quarks
in (6.9) is mimicked in the classical construction (10.1) by combined effect
of the 2 chiral ways (9.11-12) of extending the superalgebra $SU(2/1)$
curvature of [TM06] to the H-algebra.

The interesting point is that, using (6.2) and (3.6), the effective metric for
the Yang-Mill curvature $F^{a}$ comes out as

$2\;g_{ab}-(1+\alpha)\;\widehat{g}_{ab}=\hbox{diag}(1-\alpha,1-\alpha,1-\alpha,3+\alpha)\,,\qquad
a,b=1,2,3,8$ (73)

and we learn the choice of the $\alpha$ parameter in the definition of the
$\widetilde{G}$ hyper-metric corresponds to the choice of the electro-weak
angle in the classical Lagrangian according to:

$\begin{array}[]{c}tg^{2}\theta=(1-\alpha)/(3+\alpha)<=>\alpha=1-4\;sin^{2}\theta\;.\end{array}$
(74)

Substituting (7.3), we find $\alpha=-1/2$.

##  11 Discussion

The $SU(2/1)$ model appeared in 1979 as very compelling, but so far no setting
had been found that could explain how to extend the Yang-Mills quantum field
theory from gauging an internal Lie algebra symmetry to the case of a
superalgebra. We believe that the present paper represents a step in this
direction.

The original observation [N79, F79] is that, provided we grade the Fermions by
their chirality, the smallest Lie superalgebra $SU(2/1)$ exactly corresponds
to the standard model. The leptons and the quarks fit the 2 smallest
irreducible representations (3.1-2) and (4.1-4) of the superalgebra $SU(2/1)$.
The $U(1)$ hyper-charge, which is supertraceless, coincides with the original
choice of Weinberg [Wei67] who wanted to avoid a current coupled to the lepton
number, and explains the non existence of any massless charged particle. The
difficulty is to understand the role of the odd generators.

Our new idea is to take the $SU(2/1)$ matrices at face value, despite the fact
that they are not Hermitian (4.3), and to construct the whole Lagrangian by
studying the counterterms induced by the Fermions. The caveat is to remember
the triangle anomaly (4.5). It is avoided if and only if we always consider as
our building block a whole family, constituted of one lepton triplet and three
quark quadruplets, for example the electron, the left neutrino and 3 colored
copies of the up and down quarks, counting independently the left and right
chiral states. If we associate a single real scalar to each odd generator the
induced scalar propagator vanishes (4.8). But if we split the odd matrices
into their chiral parts (5.2), and consider 2 sets of scalars $\Phi_{L}$ and
$\Phi_{R}$ (5.3) the leptons and the quarks conspire to induce a the scalar
propagator (5.7) proportional to the odd part of the skew-symmetric Killing
metric of the superalgebra (3.6). It is then possible to change variables
(5.8) from $\Phi_{L},\;\Phi_{R}$ to $H,\;K$ and recover two Higgs scalar
doublets, $H$ coupled to the electron (5.12) and down quark (5.13), and $K$
coupled to the up quark (5.14). If we compute in the same way the quartic
scalar counterterms we recover (6.12) the potential first considered by Fayet
[F74], a building stone of what later became the Minimal Standard
Supersymmetric Model (MSSM). This is rather surprising, because we have not
introduced any Wess-Zumino supersymmetry in our construction, but may be
attributed to the near unicity of the $SU(2)U(1)$ quartic invariants.

Some interesting results follow from the analysis of the quantum stability of
the symmetry breaking pattern of the theory. In the Fayet potential, the
crucial element is the form of the mixed term which insures that, in the
vacuum, the 2 fields $H$ and $K$ are parallel, and therefore leave an unbroken
$U(1)$ symmetry, gauged by a massless photon. In our model, this condition is
a consequence of the super-Jacobi identity (6.10) and is stable against the
Fermionic 1-loop quantum corrections, since this is the way we derived the
scalar potential. Furthermore, if fix (7.3) the relative strength of the 2
vectors coupling to $tg^{2}\theta=5/3$ (as in $SU(5)$ grand unified theory
[GG74]), the strength of the scalar to $2/{\sqrt{3}}$, and the quartic self
coupling to $\lambda=4/9$, these coefficients are stable against the quantum
correction induced by the Fermion loops for any number of families, thanks to
the Ward identities (7.5). These identities are typical of a Yang-Mills
minimal coupling and states that scalar Fermion coupling and the scalar
quartic self coupling are indeed proportional to $g$ and $g^{2}$ where $g$ is
the Yang-Mills $SU(2)$ coupling constant. This fixes (8.7) the mass of the
neutral Higgs at
$M^{2}_{H^{0}_{1}}+M^{2}_{H^{0}_{2}}=2\;(4/3\;M_{W})^{2}=2\;(107.2\;GeV)^{2}$.
The mass of the charged Higgs corresponds to the mixed $H$ $K$ term in the
potential and gives (8.6): $M_{H^{\pm}}=2{\sqrt{2}}/3\;M_{W}=75.8\;GeV$, a
value a few $GeV$ too low relative to the minimum given by the particle data
group [Yao06]. Finally the mass of the ’$A_{0}$’ pseudo-scalar is not
constrained.

On the other hand, we clearly have a difficulty with the mass of the quarks.
If we assume that the 2 Higgs have the same mass and that the $SU(2/1)$
minimal coupling (5.3) directly gives the mass of the heaviest quark, we
predict $M_{top}=M_{H^{0}}=4/3\;M_{W}=107.2\;GeV$ which falls way under the
current experimental value $M_{top}=174.3\pm 1.8\;GeV$, deduced from direct
observations [Yao06]. But a worse problem is that the $charm$ and $up$ quarks,
in the two other families should have the same mass, when actually they are
very light ($1.25\;GeV$ and $2\;MeV$). This problem is possibly solved by the
$SU(2/1)$ indecomposable representations which have been shown by Haussling
and Scheck to correctly reproduce the Masakawa quark mixing phenomenology
[HS94]. Assuming that the masses get redistributed, we are not so far from the
relation $M^{2}_{top}+M^{2}_{charm}+M^{2}_{up}=3\;M^{2}_{H}$. This question
needs further studies.

A classical geometric interpretation of the theory is also possible. When
expressed in terms of the $\Phi_{L}$ and $\Phi_{R}$ fields, we found that the
potential involves both the symmetric $d^{a}_{ij}$ structure constants of the
superalgebra and the skew symmetric $f^{a}_{ij}$ structure constants of
$SU(3)$, times the chirality operator (3.4). This gives the idea of revisiting
the Hermitian algebras introduced by Sternberg and Wolf [SW78]. Our new result
is to show that this structure exactly fits the chiral decomposition (5.3) of
the $\Phi$ fields, allowing us to construct a classical curvature 2-forms
(9.12), where the unexpected apparition of the chirality operator $\chi$ in
the definition of the Hermitian Lie bracket (9.9) exactly compensates the
signs implied when adapting our chiral connection 1-form $\widetilde{A}$
[TM06] to the doubling of the $\Phi$ fields. There is also a probable relation
with non commutative differential geometry, as discussed in [TM06].

We are now in a position to give a classical geometrical construction of the
vector/scalar part of our Lagrangian as the square of the curvature of the
$SU(2/1)$ Hermitian algebra. In the earlier work of Ne’eman and others, the
central problem was the choice of the quadratic form needed to compute the
square of the curvature. The natural choice was the supertrace metric (3.6),
but this yields a negative propagator for the $U(1)$ field and breaks
unitarity. The phenomenological choice seemed to be the trace of the lepton
representation (3.1) which yields $sin^{2}\theta=.25$. In 1979, this value was
acceptable, but is now too high. Also the trace metric is not a good invariant
for a superalgebra, and this choice was often criticized. Our new result is to
deduce the relevant quadratic form from the renormalization theory. We found
the very elegant hyper metric $\widetilde{G}$ (6.2) which is a combination of
the trace and the supertrace metric and, thanks to the super-Jacobi identity,
contains a free parameter $\alpha$. The 2 rows and columns of $\widetilde{G}$
are used to combine the 2 $CP$ conjugated versions of the H-curvature (10.1),
and this combination restores unitarity, because in the combined Lagrangian we
can switch between the native $SU(2/1)$ geometric but non-Hermitian $\Phi_{L}$
and $\Phi_{R}$ (6.4) variables and the good Hermitian quantum field theory
scalars $H$ and $K$ (6.9). When we inject this metric in our Lagrangian
(10.1), we find that the freedom in the choice of $\alpha$ reflects a freedom
in the choice of the electro-weak angle $\alpha=1-4\;sin^{2}\theta$ (10.3) and
that the simplest choice $\alpha=0$ corresponds to the trace-metric choice,
$\alpha=-1/2$ corresponds to $sin^{2}\theta=3/8$.

Let us now recall the distinction between this model and the minimal
supersymmetric standard model (MSSM). The geometrical space behind Yang-Mills
theory is the principal fiber bundle. The base is space-time, the fiber is the
gauge group. In both cases, a superalgebra is introduced but not in the same
way. In the MSSM, the Poincare superalgebra is introduced in the base space,
acts on Bose/Fermi supermultiplets and naturally leads to supergravity. In our
approach, the superalgebra is introduced in the fiber, by embedding the
Glashow-Weinberg-Salam $SU(2)U(1)$ Lie algebra in $SU(2/1)$, it acts of
left/right Fermionic chiral supermultiplets and groups the Higgs scalars with
the Yang-Mills vectors. This construction is not incompatible with
supergravity at much higher energy or with superstrings but does not lead to
these theories. In both models, (5.11) and [F74], we are led to introduce 2
sets of scalar doublets, $H$ coupled to the electron and down-quark right
singlets, and $K$ to the up-quark singlet. The quartic scalar potential has
the same structure (6.9-12) and implies the survival of an unbroken $U(1)$ and
a massless photon. So the scalar spectrum is similar, 3 states become the
longitudinal components of the $W$ and $Z$ vector fields, 5 survive. In both
models the scale of the potential is fixed and the values are in the same
range. But of course the main difference is that the MSSM predicts for every
known particle the existence of supersymmetric partner, like the winos,
gluinos, squarks and sleptons, whereas $SU(2/1)$ fits exactly the known
particles but does not predict any new ones, except the Higgs fields. The
existence or non existence of these particles below $1\;TeV$ will allow us to
choose between the 2 kinds of supersymmetry.

The $SU(2/1)$ theory is testable, but incomplete in several directions. At the
experimental level, the model predicts 2 complex doublets of Higgs fields,
rather than 1 as in the original Weinberg paper, and constrains their masses
within the range of the new CERN collider which opens this year. At the
phenomenological level, the mixing of the quark families is not understood in
details, but may be linked to the indecomposable representations of $SU(2/1)$.
At the theoretical level, the balance between the quarks and leptons is broken
by the quantum counterterms induced by the gluons, and all the scalar
couplings, except those insuring the existence of a massless photon, are
perturbed by the Boson loops. We believe that this problem could indicate the
incompleteness of the current quantization procedure and will discuss it
elsewhere.

But even if these difficulties cannot be solved immediately, the 1-loop
stability of the structure of the scalar potential, ensuring a massless
photon, and the resulting prediction of the number and masses of the Higgs
fields seem worth reporting as a first step towards the construction of
$SU(2/1)$ Quantum Astheno Dynamics.

###### Acknowledgments.

In many ways, this paper is a concretization of our long lasting collaboration
with Yuval Ne’eman on SU(2/1). It is also a pleasure to thank Pierre Fayet and
Victor Kac for many discussions, Florian Scheck and Shlomo Sternberg for
communications, David Lipman for always prompting new ideas and Danielle
Thierry-Mieg for important suggestions.

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