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Fig. 3. Tri- and quadrilinear gauge couplings, minimal gauge coupling to fermions,
Higgs self-coupling and Yukawa coupling


In the standard model, there are 27 complex Yukawa couplings, m = 27.
The Yang“Mills and Dirac actions, contain three types of couplings, a tri-
linear self coupling AAA, a quadrilinear self coupling AAAA and the trilinear
Minimal coupling ψ — Aψ. The gauge self couplings are absent if the group G
is Abelian, the photon has no electric charge, Maxwell™s equations are linear.
The beauty of gauge invariance is that if G is simple, all these couplings are
¬xed in terms of one positive number, the gauge coupling g. To see this, take
an orthonormal basis Tb , b = 1, 2, ... dim G of the complexi¬cation gC of the Lie
algebra with respect to the invariant scalar product and an orthonormal basis
Fk , k = 1, 2, ... dim HL , of the fermionic Hilbert space, say HL , and expand the
actors,
A =: Ab Tb dxµ , ψ =: ψ k Fk . (42)
µ

Insert these expressions into the Yang“Mills and Dirac actions, then you get the
following interaction terms, see Fig. 3,
g 2 Aa Ab Ac Ad fab e fecd
g ‚ρ Aa Ab Ac fabc ρµνσ ρµνσ
, ,
µνσ µνρσ

g ψ k— Ab γ µ ψ tb k , (43)
µ

with the structure constants fab e ,
[Ta , Tb ] =: fab e Te . (44)
The indices of the structure constants are raised and lowered with the matrix of
the invariant scalar product in the basis Tb , that is the identity matrix. The tb k
is the matrix of the operator ρL (Tb ) with respect to the basis Fk . The di¬erence
˜
between the noble and the cheap actions is that the Higgs couplings, » and µ
in the standard model, and the Yukawa couplings gY j are arbitrary, are neither
connected among themselves nor connected to the gauge couplings gi .

3.3 The Winner
Physicists have spent some thirty years and billions of Swiss Francs playing on
the slot machine by Yang, Mills and Higgs. There is a winner, the standard
model of electro“weak and strong forces. Its bills are
= SU (2) — U (1) — SU (3)/(Z2 — Z3 ),
G (45)
Forces from Connes™ Geometry 297
3
HL = (2, 1 , 3) • (2, ’ 1 , 1) , (46)
6 2
1
3
HR = (1, 2 , 3) • (1, ’ 1 , 3) • (1, ’1, 1) , (47)
3 3
1
HS = (2, ’ 1 , 1), (48)
2

where (n2 , y, n3 ) denotes the tensor product of an n2 dimensional representation
of SU (2), an n3 dimensional representation of SU (3) and the one dimensional
representation of U (1) with hypercharge y: ρ(exp(iθ)) = exp(iyθ). For historical
reasons the hypercharge is an integer multiple of 1 . This is irrelevant: only the
6
product of the hypercharge with its gauge coupling is measurable and we do
not need multi-valued representations, which are characterized by non-integer,
rational hypercharges. In the direct sum, we recognize the three generations of
fermions, the quarks are SU (3) colour triplets, the leptons colour singlets. The
basis of the fermion representation space is

u c t νe νµ ν„
, , , , ,
d s b e µ „
L L L L L L



uR , cR , tR ,
eR , µR , „R
dR , sR , bR ,

The parentheses indicate isospin doublets.
The eight gauge bosons associated to su(3) are called gluons. Attention, the
U (1) is not the one of electric charge, it is called hypercharge, the electric charge
is a linear combination of hypercharge and weak isospin, parameterized by the
weak mixing angle θw to be introduced below. This mixing is necessary to give
electric charges to the W bosons. The W + and W ’ are pure isospin states, while
the Z 0 and the photon are (orthogonal) mixtures of the third isospin generator
and hypercharge.
Because of the high degree of reducibility in the bills, there are many coins,
among them 27 complex Yukawa couplings. Not all Yukawa couplings have a
physical meaning and we only remain with 18 physically signi¬cant, positive
numbers [13], three gauge couplings at energies corresponding to the Z mass,

g1 = 0.3574 ± 0.0001, g2 = 0.6518 ± 0.0003, g3 = 1.218 ± 0.01, (49)

two Higgs couplings, » and µ, and 13 positive parameters from the Yukawa
couplings. The Higgs couplings are related to the boson masses:

mW = 1 g2 v = 80.419 ± 0.056 GeV, (50)
2

g1 + g2 v = mW / cos θw = 91.1882 ± 0.0022 GeV,
12 2
mZ = (51)
2
√√
mH = 2 2 » v > 98 GeV, (52)
298 T. Sch¨cker
u

with the vacuum expectation value v := 1 µ/ » and the weak mixing angle θw
2
de¬ned by
’2 ’2 ’2
sin2 θw := g2 /(g2 + g1 ) = 0.23117 ± 0.00016. (53)

For the standard model, there is a one“to“one correspondence between the phys-
ically relevant part of the Yukawa couplings and the fermion masses and mixings,

me = 0.510998902 ± 0.000000021 MeV,
mµ = 0.105658357 ± 0.000000005 GeV,
m„ = 1.77703 ± 0.00003 GeV,


mu = 3 ± 2 MeV, md = 6 ± 3 MeV,
mc = 1.25 ± 0.1 GeV, ms = 0.125 ± 0.05 GeV,
mt = 174.3 ± 5.1 GeV, mb = 4.2 ± 0.2 GeV.

For simplicity, we take massless neutrinos. Then mixing only occurs for quarks
and is given by a unitary matrix, the Cabibbo“Kobayashi“Maskawa matrix
« 
Vud Vus Vub
CKM :=  Vcd Vcs Vcb  . (54)
Vtd Vts Vtb

For physical purposes it can be parameterized by three angles θ12 , θ23 , θ13 and
one CP violating phase δ:
« 
s13 e’iδ
c12 c13 s12 c13
 ’s12 c23 ’ c12 s23 s13 eiδ c12 c23 ’ s12 s23 s13 eiδ s23 c13  , (55)
CKM =
s12 s23 ’ c12 c23 s13 eiδ ’c12 s23 ’ s12 c23 s13 eiδ c23 c13

with ckl := cos θkl , skl := sin θkl . The absolute values of the matrix elements in
CKM are:
« 
0.9750 ± 0.0008 0.223 ± 0.004 0.004 ± 0.002
 0.222 ± 0.003 0.9742 ± 0.0008 0.040 ± 0.003  . (56)
0.009 ± 0.005 0.039 ± 0.004 0.9992 ± 0.0003

The physical meaning of the quark mixings is the following: when a su¬ciently
energetic W + decays into a u quark, this u quark is produced together with
¯
a d quark with probability |Vud |2 , together with a s quark with probability
¯
¯ quark with probability |Vub |2 . The fermion masses
|Vus | , together with a b
2

and mixings together are an entity, the fermionic mass matrix or the matrix
of Yukawa couplings multiplied by the vacuum expectation value.
Let us note six intriguing properties of the standard model.

• The gluons couple in the same way to left- and right-handed fermions, the
gluon coupling is vectorial, the strong interaction does not break parity.
Forces from Connes™ Geometry 299

• The fermionic mass matrix commutes with SU (3), the three colours of a
given quark have the same mass.
• The scalar is a colour singlet, the SU (3) part of G does not su¬er spontaneous
symmetry break down, the gluons remain massless.
• The SU (2) couples only to left-handed fermions, its coupling is chiral, the
weak interaction breaks parity maximally.
• The scalar is an isospin doublet, the SU (2) part su¬ers spontaneous sym-
metry break down, the W ± and the Z 0 are massive.
• The remaining colourless and neutral gauge boson, the photon, is massless
and couples vectorially. This is certainly the most ad-hoc feature of the stan-
dard model. Indeed the photon is a linear combination of isospin, which cou-
ples only to left-handed fermions, and of a U (1) generator, which may couple
to both chiralities. Therefore only the careful ¬ne tuning of the hypercharges
in the three input representations (46-48) can save parity conservation and
gauge invariance of electro“magnetism,

yuR = yqL ’ y ydR = yqL + y L , yeR = 2y L , y• = y L , (57)
L


The subscripts label the multiplets, qL for the left-handed quarks, L for the
left-handed leptons, uR for the right-handed up-quarks and so forth and •
for the scalar.
Nevertheless the phenomenological success of the standard model is phenome-
nal: with only a handful of parameters, it reproduces correctly some millions
of experimental numbers. Most of these numbers are measured with an accu-
racy of a few percent and they can be reproduced by classical ¬eld theory, no
needed. However, the experimental precision has become so good that quantum
corrections cannot be ignored anymore. At this point it is important to note that
the fermionic representations of the standard model are free of Yang“Mills (and
mixed) anomalies. Today the standard model stands uncontradicted.
Let us come back to our analogy between the Balmer“Rydberg formula and
the standard model. One might object that the ansatz for the spectrum, equation
(11), is completely ad hoc, while the class of all (anomaly free) s is distinguished
by perturbative renormalizability. This is true, but this property was proved [14]
only years after the electro“weak part of the standard model was published [15].
By placing the hydrogen atom in an electric or magnetic ¬eld, we know exper-
imentally that every frequency ˜state™ n, n = 1, 2, 3, ..., comes with n irreducible
unitary representations of the rotation group SO(3). These representations are
labelled by , = 0, 1, 2, ...n ’ 1, of dimensions 2 + 1. An orthonormal basis of
each representation is labelled by another integer m, m = ’ , ’ + 1, ... . This
experimental fact has motivated the credo that particles are orthonormal basis
vectors of unitary representations of compact groups. This credo is also behind
the standard model. While SO(3) has a clear geometric interpretation, we are
still looking for such an interpretation of SU (2) — U (1) — SU (3)/[Z2 — Z3 ].
We close this subsection with Iliopoulos™ joke [16] from 1976:
300 T. Sch¨cker
u

Do-It-Yourself Kit for Gauge Models:
1) Choose a gauge group G.
2) Choose the ¬elds of the “elementary particles” you want to introduce, and
their representations. Do not forget to include enough ¬elds to allow for the
Higgs mechanism.
3) Write the most general renormalizable Lagrangian invariant under G. At
this stage gauge invariance is still exact and all vector bosons are massless.
4) Choose the parameters of the Higgs scalars so that spontaneous symmetry
breaking occurs. In practice, this often means to choose a negative value
[positive in our notations] for the parameter µ2 .
5) Translate the scalars and rewrite the Lagrangian in terms of the translated
¬elds. Choose a suitable gauge and quantize the theory.
6) Look at the properties of the resulting model. If it resembles physics, even
remotely, publish it.
7) GO TO 1.
Meanwhile his joke has become experimental reality.

3.4 Wick Rotation
Euclidean signature is technically easier to handle than Minkowskian. What is
more, in Connes™ geometry it will be vital that the spinors form a Hilbert space
with a true scalar product and that the Dirac action takes the form of a scalar
product. We therefore put together the Einstein“Hilbert and Yang“Mills“Higgs
actions with emphasis on the relative signs and indicate the changes necessary
to pass from Minkowskian to Euclidean signature.
In 1983 the meter disappeared as fundamental unit of science and technology.
The conceptual revolution of general relativity, the abandon of length in favour
of time, had made its way up to the domain of technology. Said di¬erently,
general relativity is not really geo-metry, but chrono-metry. Hence our choice of
Minkowskian signature is + ’ ’’.
With this choice the combined Lagrangian reads,

{’ 16πG ’ 16πG R ’ 2g2 tr (Fµν F µν ) + g12 m2 tr (A— Aµ )

2Λc 1 1
A µ
—µ
+ 2 (Dµ •) D • ’ 2 m• |•| + 2 µ |•| ’ »|•|
2 2 12 2 4
1 1

+ ψ — γ 0 [iγ µ Dµ ’ mψ 14 ] ψ} |det g·· |1/2 . (58)

This Lagrangian is real if we suppose that all ¬elds vanish at in¬nity. The relative
coe¬cients between kinetic terms and mass terms are chosen as to reproduce the
correct energy momentum relations from the free ¬eld equations using Fourier
transform and the de Broglie relations as explained after equation (34). With
the chiral decomposition
1’γ5 1+γ5
ψL = ψ, ψR = ψ, (59)
2 2

the Dirac Lagrangian reads
Forces from Connes™ Geometry 301

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