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