Noncommutative Geometry constraints and the Standard ...present the renormalization group (RG) analysis and show that the corresponding evo-lution diﬀerential equations, once solved

EJTP 5, No. 17 (2008) 41–64 Electronic Journal of Theoretical Physics

Noncommutative Geometry constraints and theStandard Renormalization Group approach: Two

Doublets Higgs Model as an Example

N.Mebarki and M.Harrat

Laboratoire de Physique Mathématique et Subatomique,Mentouri University, Constantine, Algeria

Received 1 June 2007, Accepted 26 October 2007, Published 27 March 2008

Abstract: The Chamssedine-Fröhlich Approach to Noncommutative Geometry (NCG) isextended and applied to the reformulation of the two doublets Higgs model. The Fuzzy mass,coupling and unitarity relations are derived. It is shown that the latter are no more preservedunder the renormalization group equations obtained from the standard quantization method.This suggests to look for an appropriate NCG quantization procedure.c© Electronic Journal of Theoretical Physics. All rights reserved.

Keywords: Noncommutative Geometry; Renormalization Group; Higgs ModelPACS (2006): 02.40.k; 02.40.Gh; 14.80.Bn; 13.66.Fg

1. Introduction

In the last decade many interest has been devoted to understand the unsolved problems

of the standard model and to find some origins and mecanisms of the various parameters.

Many approachs have beeen developped namely the Connes Noncommutative geometry

[1] − [20] formalism. The latter is based on a unitary algebra, where gauge bosons andHiggs are treated in an equal footing. The drawback of this approach, is that we cannot

go beyond the standard model[1] − [3]. To overcome this situatio, Chamessedine andFröhlich proposed an extension of the Dirac operator which allows to consider other

unified models based on unitary and orthogonal groups SU(5), SO(10), ..etc.[21] − [23].

42 Electronic Journal of Theoretical Physics 5, No. 17 (2008) 41–64

However, the spontaneous breakdown of the symmetry is not well controled to allow

getting reasonable mass and mixing angles relations. Moreover, the elimination of the

auxilliary unphysical fields undergo through field equations which is in general a com-

plicated procedure[21] − [23]. The goal of this paper is two fold: First to extend theChamesseddine - Frohlich approach (through the two doublet Higgs model) by includ-

ing the strong interaction sector in the new mathematical formalism and generalizing

the Dirac operator as well as the scalar product. The elimination of the junks forms

is done by applying the orthogonality condition before the construction of the action.

Second to derive the tree level Fuzzy mass, coupling and unitarity relations and to show

that we cannot preserve these constraints under the ordinary renormalization group flow

(running masses, couplings etc..).

In section 2, we give the formalism and derive some of the NCG constraints.We

present the renormalization group (RG) analysis and show that the corresponding evo-

lution differential equations, once solved will not in general preserve the tree level NCG

relations. Finally, in section 3, we draw our conclusions.

2. Formalism

Consider a model consisting of the spectral triplet (A,H,D) where A and D are aninvolutive algebra of operators and unbounded self-adjoint operators on a Hilbert space

H respectively. Let X be a compact Riemanian spin-manifold, and (A1,H1,D1, χ) theDirac K- cycle where H1 ≡ L2(X,√gd4x) (g is the metric) acts on the A1 algebra offunctions on the X manifold. Let (A2,H2,D2) another triplet where A2 is given by[1] , [2] , [3] , [7]:

A2 = M2(C) ⊕ C ⊕ M3(C) (1)where M2(C) ,(resp.M3(C)) are 2x2 ,(resp.3x3 ) matrices algebra and H2 = h2,1⊕ h2,2 ⊕h2,3 with h2,1, h2,2 and h2,3 are Hilbert spaces on C

2, C and C3 respectively. Then, Aand D are taken to be:

A = A1 ⊗A2 (2)

H = H1 ⊗H2 (3)and

Electronic Journal of Theoretical Physics 5, No. 17 (2008) 41–64 43

D = D1 ⊗ 1 ⊕ χ ⊗D2 (4)Where χ is chirality operator χ acting on the Hilbert space H such that:

χ = χ∗, χ2 = 1, χD = −Dχ (5)In this case, the Hilbert space H can be splitted into two orthogonal subspaces HL and

HR such that:H = HL ⊕HR (6)

and

HL,R = 12(1 ∓ χ)H (7)

or in the formal form:

HL =(C

2 ⊗ CN ⊗ C3)⊕ (C2 ⊗ CN ⊗ C) (8)and

HR =((C + C) ⊗ CN ⊗ C3)⊕ (C ⊗ CN ⊗ C) (9)

Therefore, the A algebra which represents the gauge group U(2) ⊗ U(1) ⊗ U(3) can bewritten in the following form:

A = C∞ (X) ⊗ (M2 (C) ⊕ C ⊕ M3 (C)) (10)

where here, C∞ (X) is the algebra of differentiable functions. It is to be noted that, aninvolutive representation π of this algebra is provided by the map:

π : A → B(H) (11)f → 14x4 ⊗ diag(f1, f2, f3, f4, f5, f6) (12)

where B(H) is the algebra of bounded operators in the Hilbert space H and to every blockdiagonal f ∈ A, we associate a sixtet (f1, f2, f3, f4, f5, f6) of matrix-valued functions onX such that:


fi ∈ M2(C); i = 1, 2 (13)fi ∈ C; i = 3, 4 (14)fi ∈ M3(C); i = 5, 6 (15)

Now, Regarding the explicit form of the Hermitian Dirac operator D, it is givenby:

D =

⎛⎜⎜⎜⎜⎜⎜⎜⎝

∂ ⊗ 1 ⊗ 1N γ5 ⊗ M12 ⊗ K12 ... γ5 ⊗ M16 ⊗ K16γ5 ⊗ M21 ⊗ K21 ∂ ⊗ 1 ⊗ 1N ... γ5 ⊗ M26 ⊗ K26

... ... ... ...

γ5 ⊗ M61 ⊗ K61 γ5 ⊗ M62 ⊗ K62 ... ∂ ⊗ 1 ⊗ 1N

⎞⎟⎟⎟⎟⎟⎟⎟⎠(16)

where ∂ = γμ∂μ. The reality condition of the operator D implies that:

M∗mn = Mnm, K∗mn = Knm, m = n = 1, 2, ..6 (17)

The Mmn matrices determine the tree level vacuum expectation values of the higgs

fields and control the corresponding symmetry breaking scheme. The NxN, Kij (i, j =

1, 6) matrices where N is the fermionic family number; are introduced to include

informations about the mixing between the various generations of quarks and leptons.

Now, in order to get the standard model with two Higgs doublets, we have to choose:

K13 = K24 = K, (18)

K31 = K42 = K∗

and take:

M13 =

⎛⎜⎝ M̃uM̃d

⎞⎟⎠ = S ′ ⊗ M̃u ⊗ 1N + S ⊗ M̃d ⊗ 1N (19)and

M24 =

⎛⎜⎝ 0M̃e

⎞⎟⎠ = S ⊗ M̃e ⊗ 1N (20)


where

S =

⎛⎜⎝ 01

⎞⎟⎠ (21)and

S ′ =

⎛⎜⎝ 10

⎞⎟⎠ (22)The remaining Kmn and Mmn matrices vanish. The M̃u, M̃d and M̃e are the Up, Down

quarks-like and leptons mass matrices respectively such that:

M̃u =

⎛⎜⎜⎜⎜⎝mu

0

0

0

mc

0

0

0

mt

⎞⎟⎟⎟⎟⎠ (23)

M̃d = CCKM

⎛⎜⎜⎜⎜⎝md

0

0

0

ms

0

0

0

mb

⎞⎟⎟⎟⎟⎠ (24)and

M̃e =

⎛⎜⎜⎜⎜⎝me

0

0

0

mμ

0

0

0

mτ

⎞⎟⎟⎟⎟⎠ (25)where

CCKM =

⎛⎜⎜⎜⎜⎝Vud

Vcd

Vtb

Vus

Vcs

Vts

Vub

Vcb

Vtb

⎞⎟⎟⎟⎟⎠ (26)


is the Cabbibo-Kobayashi-Maskawa matrix which is supposed non degenerate and the

elements ml, (l = u, c, t, d, s, b, e, μ , τ ) are positive fermions masses).

If we denote by Ωp (A) the algebra of p-forms where Ω0 (A) = A, the ele-ments of the canonical universal algebra ΩD (A) can be obtained from those of A asfollows:

ΩD (A) := ⊕p≥0

ΩpD (A) =π (Ω (A))π (Aux)

(27)

where:

ΩpD (A) =(

π(Ωp(A))π(Auxp)

)(28)

represents the p-form canonical differential algebra and Auxp are the p junks forms (

non dynamical auxilliary fields). It is worth to mention that the scalar product between

the elements of this algebra is defined as:

ΩpD (A) ⊗ ΩlD (A) → IR (29)(ω, η) → 〈ω, η〉 = δplTr (ω∗ηZ)

Here Tr stands for trace and the positive definite operator Z has as expression:

Z =

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

x 0 0 0 0 0

0 x 0 0 0 0

0 0 y 0 0 0

0 0 0 y 0 0

0 0 0 0 z 0

0 0 0 0 0 z

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠(30)

with

x → 14 ⊗ 12 ⊗ x ⊗ 1Nx → 14 ⊗ 11 ⊗ y ⊗ 1N (31)z → 14 ⊗ 13 ⊗ z ⊗ 1N

( x , y and z ∈ M3(C) and 1D are DxD identity matrices). Notice that Z verifies thefollowing commutation relations:


[D, Z] = 0 (32)and

[Z, π (A)] = 0 (33)which corresponds to the case:

x = x.13 , y = y.13 , z = z.13 (34)

with x, y and z ∈ R. Now, the 1-form differential algebra Ω1D (A) reads:

Ω1D (A) ≈ π(Ω1 (A)) = {π (ω) = π(∑

1

αiδβi

)=∑

1

π(αi) [D, π (βi)]} (35)

where αi, βi ∈ A. If we denote by ωmμ and Φmn, (m = n = 1, 2, ..6 ) the vector and scalarfields respectively and Nf the fermionic family number (in our case Nf = 3) such that:

(ωmμ )∗ = −ωmμ (36)

and

Φ∗mn = Φmn (37)

Then, the non vanishing elements are given by:

π (ω)ii = γi ⊗ ωiμ ⊗ 13 ⊗ 1N , i =

−→1, 6

π (ω)13 = γ5 ⊗ Φ13 ⊗ K13 (38)π (ω)31 = γ5 ⊗ Φ31 ⊗ K31π (ω)24 = γ5 ⊗ Φ24 ⊗ K24

and

π (ω)42 = γ5 ⊗ Φ42 ⊗ K42 (39)with


Φ13 =

⎛⎜⎝ ϕ1ϕ2 − 1

⎞⎟⎠⊗ Md ⊗ 13 +⎛⎜⎝ χ1

κ2 − 1

⎞⎟⎠⊗ Mu ⊗ 13 (40)and

Φ24 =

⎛⎜⎝ ϕ1ϕ2 − 1

⎞⎟⎠⊗ Me ⊗ 11 (41)Regarding the 2-form canonical differential algebra Ω2D (A) , it is defined as:

Ω2D (A) =π (Ω2 (A))

Aux2(42)

Where, the 2- form non physical (junks) differential algebra Aux2 can be obtained as:

Aux2 = {π (δω) /π (ω) = 0} (43)which implies that:

ωmμ = 0, m = 1, 2, ..6 (44)

and

Φmn = 0, m = n = 1, 2, ..6 (45)Straightforward calculations using the fact that:

π (δω) =

{π(δαi δβ

i)

=∑

i

[D, π (αi)] [D, π (βi)]} (46)as well as the relations:

| M31 |2= 12 ⊗ (μ̃u + μ̃d) ⊗ 13

| M13 |2= 12 ⊗ (μu + μd)13 + τ3 ⊗ 12(μu − μd) ⊗ 13 + τ1 ⊗ μud ⊗ 13

| M42 |2= 12 ⊗ μ̃e ⊗ 11 (47)


and

| M24 |2= (12 − τ3) ⊗ 12μe ⊗ 11

where

μa = M̃a.M̃∗a

μ̃a = M̃∗a .M̃a

μab = M̃a.M̃∗b (48)

and

μ̃ab = M̃∗b .M̃a

( τ3 and τ1 are the Pauli matrices) show that the elements of this algebra have as

expressions:

(Aux2

)11

= γμγν ⊗⊗13 ⊗ 1N + 14 ⊗ iκ2 ⊗ 12

(μd − μu) ⊗ 13 ⊗ KK∗+14 ⊗ iκ3 ⊗ μud ⊗ 13 ⊗ KK∗(

Aux2)22

= γμγν ⊗ κ1μν ⊗ 13 ⊗ 1N + 14 ⊗ iκ2 ⊗ μe ⊗ 11 ⊗ KK∗ (49)(Aux2

)33

=(Aux2

)44

= γμγν ⊗ κμν ⊗ 13 ⊗ 1N(Aux2

)55

=(Aux2

)66

= γμγν ⊗ yμν ⊗ 13 ⊗ 1Nand

(Aux2

)mn

= 0, m = n = 1, 6with

κ1μν = −∑

i

αi1∂μ∂νβi1

κμν = −∑

i

αi3∂μ∂νβi3

κ2 = −∑

i

αi1[i τ3, β

i1

](50)


κ3 =∑

i

αi1[i τ1, β

i1

]and

yμν = −∑

i

αi5∂μ∂νβi5 (51)

Now, taking into account the fact that the elements of π (Ω2 (A)) are given by:

π(Ω2 (A)) = {π (C) = π (δω) + π (ω2) , ω ∈ Ω1 (A)} (52)

the orthogonality condition leads to the following canonical differential algebra Ω2D (A)elements:

π (C)11 =1

2γμν ⊗ F 1μν ⊗ 13 ⊗ 1N

π (C)22 =1

2γμν ⊗ F 2μν ⊗ 13 ⊗ 1N

π (C)33 =1

2γμν ⊗ F 3μν ⊗ 13 ⊗ 1N+14 ⊗ (ϕ∗ϕ − 1) ⊗ μ̃d ⊗ 13 ⊗ K∗K + 14 ⊗ (κ∗κ − 1) ⊗ μ̃u ⊗ 13 ⊗ K∗K+14 ⊗ ϕ∗κ ⊗ μdu ⊗ 13 ⊗ K∗K + 14 ⊗ κ∗ϕ ⊗ μ̃du ⊗ 13 ⊗ K∗K

+γμγν ⊗ TrK∗K

6N

⎧⎪⎨⎪⎩ tr (3μd + μe) (ϕ∗ϕ − 1)μν + tr (3μu) (κ∗κ − 1)μν

+tr (3μdu) (ϕ∗κ)μν + tr (3μ̃du) (κ

∗ϕ)μν

⎫⎪⎬⎪⎭(53)π (C)44 =

1

2γμν ⊗ F 4μν ⊗ 13 ⊗ 1N + 14 ⊗ (ϕ∗ϕ − 1) ⊗

∼

μe ⊗ 11 ⊗ K∗K

+γμγν ⊗ TrK∗K

6N

⎧⎪⎨⎪⎩ tr (3μd + μe) (ϕ∗ϕ − 1)μν + tr (3μu) (κ∗κ − 1)μν

+tr (3μdu) (ϕ∗κ)μν + tr (3μ̃du) (κ

∗ϕ)μν

⎫⎪⎬⎪⎭π (C)55 = π (c)66 =

1

2γμν ⊗ F 5μν ⊗ 13 ⊗ 1N

π (C)13 = −γ5γμ ⊗ Dμϕ ⊗ Md ⊗ 13 ⊗ K − γ5γμ ⊗ Dμκ ⊗ Mu ⊗ 13 ⊗ Kπ (C)31 = −γ5γμ ⊗ (Dμϕ)∗ ⊗ M∗d ⊗ 13 ⊗ K∗ − γ5γμ ⊗ (Dμκ)∗ ⊗ M∗u ⊗ 13 ⊗ K∗π (C)24 = −γ5γμ ⊗ Dμϕ ⊗ Me ⊗ 11 ⊗ K

and


π (C)42 = −γ5γμ ⊗ (Dμϕ)∗ ⊗ M∗e ⊗ 11 ⊗ K∗ (54)

where

ϕ =

(ϕ1ϕ2

)(55)

χ =

(χ1χ2

)(56)

and

Dμ. = ∂μ. + ω1μ. − ω3μ (57)

The F iμν ’s are the gauge fields strength tensor.

Before we proceed with further details of the calculation, It is worth to mention that

our approach is considered to be an extension of Chamssedine-Frohlich formalism and

not that of Conne’s in the sense that like in ours, the former introduce the Mij and Kijmatrices where in the later are taken to be proportional to the identity matrix (special

case). Moreover, the difference between our approach and that of Chamssedine-Frohlich

[20] − [22] lies on:a) The generalization of the scalar product

b) The elimination of the junks forms (or the non dynamical auxilliary fields) by

using the orthogonality condition instead of the field equations which is in general a very

complicated procedure.

c) The modification of the algebra A (see Eq.(10)) for the gauge group U(2)⊗U(1)⊗U(3) instead of C∞ (X)⊗(M2 (C)⊕C) for U(2)⊗U(1) in the order to include the stronginteractions.

d) The modification and extension of the Dirac operator to have the general form

given by Eq.(16). In the case of the standard model and to account for the strong inter-

actions, quark masses and mixings the Dirac operator the following simplified expression:


D=

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

∂⊗1⊗1N 0 γ5⊗M13⊗K13 0 0 0

0 ∂⊗1⊗1N 0 γ5⊗M24⊗K24 0 0

γ5⊗M31⊗K31 0 ∂⊗1⊗1N 0 0 0

0 γ5⊗M42⊗K42 0 ∂⊗1⊗1N 0 0

0 0 0 0 ∂⊗1⊗1N 0

0 0 0 0 0 ∂⊗1⊗1N

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(58)

Notice that, in order to insure that the gauge group SU(3) remains unbroken, the mass

matrix along the M3 (C) direction is taken to be zero, forcing the vanishing of the scalar

field expectation value along the same direction. In the Chamssedine-Frohlich approach,

the general form of the Dirac operator has the following general form:

D =

⎛⎜⎜⎜⎜⎝∂ ⊗ 1 ⊗ 1N γ5 ⊗ M12 ⊗ K12 γ5 ⊗ M13 ⊗ K13

γ5 ⊗ M21 ⊗ K21 ∂ ⊗ 1 ⊗ 1N γ5 ⊗ M23 ⊗ K23γ5 ⊗ M31 ⊗ K31 γ5 ⊗ M32 ⊗ K32 ∂ ⊗ 1 ⊗ 1N

⎞⎟⎟⎟⎟⎠ (59)and in the case of the standard model it becomes (M13 = M23 = 0):

D =

⎛⎜⎜⎜⎜⎝∂ ⊗ 1 ⊗ 1N γ5 ⊗ M12 ⊗ K12 0

γ5 ⊗ M21 ⊗ K21 ∂ ⊗ 1 ⊗ 1N 00 0 ∂ ⊗ 1 ⊗ 1N

⎞⎟⎟⎟⎟⎠ (60)Notice that, in our extended approach, the matrices M13 and M24 contain the quarks and

leptons mass respectively. Similarly for the matrices K13 and K24 where they include

the quarks and leptons mixings. However, for the Chamssedine-Frohlich approach, we

have just the matrices M12 and K12 and therefore, we cannot include in a natural way

the quarks and their mixings. Furthermore, even if we try to include in their approach

the SU(3) gauge group in a commutative way and decoupling it from the rest, we will

face the quark mass problem.

In what follows, we will concentrate on the scalar fields sector. Starting from the

definition of the bosonic action ̃B:

̃B =∫

d4x Tr (trC∗C) (61)


and after diagonalization, by introducing a mixing angle ϑ such that:

tan ϑ =2tr3μdu

tr (3 (μd − μu) + μe) ≈2mtmb

m2t − m2b(62)

and spontaneous breaking of the gauge symmetry as:

U (2)L ⊗ U (1) ⊗ U (3) 〈ϕ〉 , 〈κ〉−−−−−→ U (1) ⊗ U (1)′ ⊗ U (3) (63)

where ϕ and κ are two complex doublets where:

ϕ =

⎛⎜⎝ϕ1ϕ2

⎞⎟⎠ ≡⎛⎜⎝ Φ+1

η1 + i κ1

⎞⎟⎠ (64)and

κ =

⎛⎜⎝κ1κ2

⎞⎟⎠ ≡⎛⎜⎝ Φ+2

η2 + i κ2

⎞⎟⎠The corresponding vaccum expectation values 〈ϕ〉 and 〈κ〉 are choosen as:

〈ϕ〉 =

⎛⎜⎝ 0〈ϕ0〉

⎞⎟⎠ , 〈κ〉 =⎛⎜⎝ 0

〈κ0〉

⎞⎟⎠ (65)By redefining the scalar fields ϕ and κ as:

ϕ → α1ϕ (66)

and

κ → α2κwith

α1,2 =L̃

mt cos ϑ ± mb sin ϑ (67)

where


L̃ =1

[3 (x + y) TrK∗K]1/2(68)

A lengthy calculation leads in the Higgs sector, to the following expression of the potential

V (ϕ, κ) :

V (ϕ, κ) = ξ1(ϕ∗ϕ − 〈ϕ0〉2

)2+ ξ2

(κ

∗κ − 〈κ0〉2

)2+ ξ3 (ϕ

∗ϕ) + ξ4 (κ∗κ)

+ξ5 (ϕ∗ϕ) (κ∗κ) + ξ6 (ϕ∗ϕ) (ϕ∗κ + κ∗ϕ) (69)

+ξ7 (κ∗κ) (ϕ∗κ + κ∗ϕ) + ξ8 (ϕ∗κ + κ∗ϕ)

2

+ξ9 (ϕ∗κ + κ∗ϕ)

where

ξ1 = 3yΣ(m4t cos

4 ϑ + m4b sin4 ϑ + 2m2t m

2b cos

2 ϑ sin2 ϑ + 2m3t mb sin 2ϑ + 2m3bmt sin

2 2ϑ)(α1)

4

ξ2 = 3yΣ(m4t sin

4 ϑ + m4b cos4 ϑ + 2m2t m

2b cos

2 ϑ sin2 ϑ − 2m3t mb sin 2ϑ + 2m3bmt sin2 2ϑ)(α2)

4

ξ3 = 6yΣ(m4t cos

4 ϑ + m4b sin4 ϑ + m2t m

2b − 2m3t mb sin 2ϑ

)(α1)

2

ξ4 = 6yΣ(m4t sin

4 ϑ + m4b cos4 ϑ + m2t m

2b − 2m3t mb sin 2ϑ

)(α2)

2 (70)

ξ5 = 6yΣ((

m4t + m4b

)cos2 ϑ sin2 ϑ + 2m2t m

2b

(cos4 ϑ + sin4 ϑ

)− 4m3bmt sin2 2ϑ) (α1)2 (α2)2ξ6 = 3yΣ

⎛⎜⎜⎜⎜⎝sin 2ϑ

(−m4t cos2 ϑ + m4b sin2 ϑ)+m2t m

2b cos 2ϑ sin 2ϑ + 2m

3t mb(cos

2 ϑ cos 2ϑ + 12sin2 2ϑ)+

+2m3bmt(sin2 ϑ cos 2ϑ + 1

2sin2 2ϑ)

⎞⎟⎟⎟⎟⎠ (α1)3 (α2)

ξ7 = 3yΣ

⎛⎜⎜⎜⎜⎝sin 2ϑ

(−m4t sin2 ϑ + m4b cos2 ϑ)− m2t m2b cos 2ϑ sin 2ϑ+2m3t mb(cos 2ϑ sin

2 ϑ + 12sin2 2ϑ)+

+2m3bmt(cos 2ϑ cos2 ϑ + 1

2sin2 2ϑ

⎞⎟⎟⎟⎟⎠ (α1) (α2)3

ξ8 = 3yΣ

(1

4

(m2t + m

2b

)2sin2 2ϑ + 2m3bmt cos

2 2ϑ

)(α1)

2 (α2)2

and

ξ9 = 3yΣ((

m2b − m2t)2

sin 2ϑ + 4m3t mb cos 2ϑ)

(α1) (α2)

It is easy now, to obtain the following mass matrices for the real neutral, imaginary

neutral and charged scalar fields (η1, η2) , (κ1, κ2) and(Φ±1 , Φ

±2

)respectively :


M2 (η1, η2) =

⎛⎜⎝Δ11 Δ12Δ21 Δ22

⎞⎟⎠M2 (κ1, κ2) =

⎛⎜⎝ Δ̃11 Δ̃12Δ̃21 Δ̃22

⎞⎟⎠ (71)M2

(Φ±1 , Φ

±2

)=

⎛⎜⎝ Δ̂11 Δ̂12Δ̂21 Δ̂22

⎞⎟⎠where

Δ11 = 6ξ1 〈ϕ0〉2 + (ξ5 + 4ξ8) 〈κ0〉2 + 6ξ6 〈ϕ0〉〈κ0〉 + ξ3

Δ12 = Δ21 = 3ξ6 〈ϕ0〉2 + 3ξ7 〈κ0〉2 + 2(ξ5 + 2ξ8) 〈ϕ0〉〈κ0〉 + ξ9

Δ22 = (ξ5 + 4ξ8) 〈ϕ0〉2 + 6ξ2 〈κ0〉2 + 6ξ7 〈ϕ0〉〈κ0〉 + ξ4

Δ̃11 = 2ξ1 〈ϕ0〉2 + ξ5 〈κ0〉2 + 2ξ6 〈ϕ0〉〈κ0〉 + ξ3

Δ̃12 = Δ̃21 = ξ6 〈ϕ0〉2 + ξ7 〈κ0〉2 + 2ξ8 〈ϕ0〉〈κ0〉 + ξ9

Δ̃22 = 2ξ1 〈ϕ0〉2 + ξ5 〈κ0〉2 + 2ξ7 〈ϕ0〉〈κ0〉 + ξ4

Δ̂11 = ξ3/2 + ξ1 〈ϕ0〉2 + ξ5 〈κ0〉2 /2 + ξ6 〈ϕ0〉〈κ0〉

Δ̂12 = Δ̂21 = ξ9/2 + +ξ6 〈ϕ0〉2 + ξ7 〈κ0〉2 + 2ξ8 〈ϕ0〉〈κ0〉and

Δ̂22 = ξ5 〈ϕ0〉2 /2 + ξ2 〈κ0〉2 + 2ξ7 〈ϕ0〉〈κ0〉 + ξ4/2After a diagonalisation of the above symmetric matrices (Eqs.71), and choosing ϑ = π

4,

it is easy to show that:


mt ≈ 2.4mb (72)g = (12Nfx)

−1/2 (73)

g′ =(12Nf

(x4

+ y))−1/2

(74)

and

g3 = (12Nfz)−1/2

where g, g′ and g3 are the coupling constants of the gauge groups U (2) , U (1) , U (3)respectively.Similarly,

ξ1 ≈ 0.17Σx (TrKK∗)2

,

ξ2 ≈ 0.30Σx (TrKK∗)2

,

ξ3 ≈ 0.30m2t Σ

TrKK∗,

ξ4 ≈ 1.50m2t Σ

TrKK∗, (75)

ξ5 ≈ ξ1,ξ6 = ξ7 = 0,

ξ8 ≈ 0.20Σx (TrKK∗)2

,

ξ9 ≈ 2.m2t Σ

TrKK∗

Σ = Tr (KK∗)2 − (3N)−1 (TrKK∗)2 (76)and

〈ϕ0〉〈κ0〉 � 70 (77)

which means that 〈κ0〉 � 〈ϕ0〉 . Straightforward calculations give


〈ϕ0〉2 ≈ 9x TrKK∗m2t (78)and if we take Mt ≈ 2MW , we deduce that:√

TrKK∗

x≈ 12 (79)

Regarding the Higgs masses, we obtain first from the eigenstates of Eq.(71); the

CP-odd pseudoscalar A0, CP-even scalars H0 and h0and charged scalars H± related tothe state systems (η1, η2) , (κ1, κ2) and

(Φ±1 , Φ

±2

)respectively, then we obtain in terms

of the top quark mass the following approximate mass eigenvalues relations:

M2A0 = M2H± ≈

2.64Σ

x12 (TrKK∗)

32

M2t ,

M2H0 ≈3.15Σ

x12 (TrKK∗)

32

M2t (80)

M2h0 ≈0.42Σ

x12 (TrKK∗)

32

M2t

with:

M2t = c2 (mt 〈ϕ0〉 + mb 〈κ0〉)

Now, using the Schwartz inequality, one can deduce that:

Σ ≤ 269

(TrK∗K)2 (81)

and therefore, we get for the CP-odd pseudoscalar A0 and charged scalars H± the con-straint:

MA0 = MH± � 9.57Mt (82)

Similarly, for the CP-even scalars H0 and h0one can show that:

MH0 � 10.45Mt (83)

and

Mh0 � 3.81Mt (84)


Notice that for the MH± and MH0 , our results are compatible with the actual experi-

mental bounds which allow for a heavy Higgs scenario.

Regarding the tree level NCG unitarity bound, we have to consider the scalar elastic

and inelastic processes of the form B1 + B2 → B3 + B4 by using the partial wavedecomposition technique to the corresponding amplitude M :

M(s, t, u) = 16π∞∑l=0

(2l + 1)Pl(cos θ)Al(s) (85)

To get upper bound limits to the scalar potential parameters of Eq.(75), one has

to require that the tree level unitarity has to be preserved in all possible scattering

processes. This is equivalent to the fact that the s partial wave amplitude A0 for the

scalar-scalar, gauge boson- gauge boson and gauge boson-scalar has to satisfy |A0| ≤ 12 inthe high energy limit. We remind the reader that at a very high energy, the equivalence

theorem states that the amplitude of a scattering process involving longitudinal gauge

bosons W±μ and Zμ may be approximated by the scalar amplitude in which gauge bosonsare replaced by their corresponding Goldostone bosons and the unitarity constraints

can be implemented by only considering the pure scalar scatterings. In what follows,

we limit ourselves to pure scalar scattering processes dominated by quartic interactions

and follow the technique introduced in ref.[25]. The S matrix expressed in terms of the

physical fields can be transformed into an S matrix for the non physical fields w±j , hjand zj (i = 1, 2) such that :

ϕ =

⎛⎜⎝ w+1〈ϕ0〉” + 12(h1 + z1)

⎞⎟⎠ ,κ =

⎛⎜⎝ w+2〈κ0〉” + 12(h2 + z2)

⎞⎟⎠ (86)and

w±j = α−1j Φ

±j ; hj = 2α

−1j ηj; zi = 2α

−1j κj; 〈ϕ0〉” = α−11 〈ϕ0〉 ; 〈κ0〉” = α−12 〈κ0〉

(87)

by making a unitary transformation. Thus, the full set of the scalar scattering processes

can be expressed as an S matrix 22x22 composed of 4 submatrices M16X6,M26X6,M

32X2

and M48X8 which do not couple with each other and where the entries are the quartic


couplings mediating the scattering processes. In our case, straightforward simplifications

lead to:

(a)Basis (w+1 w−2 , w

+2 w

−1 , h1z2, h2z1, z1z2, h1h2):

M16X6 =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

ξ5 − 14ξ8 ξ8 0 0 ξ8 ξ8ξ8 ξ5 +

14ξ8 0 0 ξ8 ξ8

0 0 ξ5ξ82

0 0

0 0 ξ82

ξ5 0 0

ξ82

ξ82

0 0 ξ5 + ξ8ξ82

ξ82

ξ82

0 0 ξ82

ξ5 + ξ8

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦(88)

(b)Basis (w+1 w−1 , w

+2 w

−2 ,

z1z1√2, z2z2√

2, h1h1√

2, h2h2√

2):

M26X6 =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

4ξ1 ξ5 +12ξ8

√2ξ1

1√2ξ5

√2ξ1

1√2ξ5

ξ5 +12ξ8 4ξ2

1√2ξ5

√2ξ2

1√2ξ5

√2ξ2

√2ξ1

1√2ξ5 3ξ1 ξ5 +

12ξ8 ξ1

12ξ5

1√2ξ5

√2ξ2 ξ5 +

12ξ8 3ξ2

12ξ5 ξ2

√2ξ1

1√2ξ5 ξ1

12ξ5 3ξ1 ξ5 +

12ξ8

1√2ξ5

√2ξ2

12ξ5 ξ2 ξ5 +

12ξ8 3ξ2

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦(89)

(c)Basis (h1z1, h2z2):

M32X2 =

⎡⎢⎣ 2ξ1 12ξ812ξ8 2ξ2

⎤⎥⎦ (90)(d)Basis (h1w

+1 , h2w

+1 , z1w

+1 , z2w

+1 , h1w

+2 , h2w

+2 , z1w

+2 , z2w

+2 ):


M48X8 =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

2ξ1 0 0 0 012ξ8 0 0

0 ξ5 0 012ξ8 0 0 0

0 0 2ξ1 0 0 0 0 0

0 0 0 ξ5 0 012ξ8 0

0 12ξ8 0 0 ξ5 0 0 0

12ξ8 0 0 0 0 2ξ2 0 0

0 0 0 12ξ8 0 0 ξ5 0

0 0 12ξ8 0 0 0 0 2ξ2

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(91)

Now, using Mathematica one can diagonalize the Mi (i = 1, 4) matrices to get the

following eigensvalues:

Ω11 = Ω12 = ξ5 −

1

2ξ8, Ω

13 = ξ5 +

5

2ξ8, Ω

14 = Ω

15 = ξ5 +

1

2ξ8 (92)

Ω21,2 = 3(ξ1 + ξ2)±√

9(ξ1 − ξ2)2 + (2ξ5 + ξ8)2, Ω23,4 = Ω25,6 = ξ1 + ξ2 ±√

(ξ1 − ξ2)2 + 14ξ28

(93)

Ω31,2 = Ω23,4 (94)

and

Ω41 = Ω14, Ω

42 = Ω

12, Ω

43 = Ω

15, Ω

44,5 = Ω

25,6, Ω

46,7 = Ω

24,6, Ω

41 = Ω

42 (95)

where Ωji stands for the ith eigenvalue of the matrices M j. Now, imposing the unitarity

condition

∣∣Ωji ∣∣ ≤ 8π (96)and replacing the ξi’s by their expressions (Eqs.75), we end up with the following solution:

Σ

(TrKK∗)2� 8πx

14, 6(97)


which can be considered as a new unitarity constraints between the various NCG pa-

rameters. Notice that in order that Eq.(97) is compatible with the Schwartz inequality

(Eq.(81)), one has to have x � 1.7.

Regarding the quantization, in the framework of NCG, there is no satisfactory pro-

cedure which has been developed yet treating the gauge and higgs bosons in an equal

footing. Therefore, we expect that the quantum fluctuations may badly violate the tree

level NCG constraints and relations. In principle, the change in the quantization rules is

needed around certain energy scale and we have to assume that just below such a scale,

the standard quantization method makes a good approximation. In this case, we can

start from the classical lagrangian and use the conventional quantization.

In what follow, we apply this approach to our NCG approach of the two doublets

Higgs model through simple examples and test whether it is possible to preserve the tree

level mass, coupling and unitarity relations. let us take for example the ratio ξ8g.It is

easy to see from the relations Eqs.(73)-(74) and Eq.(75) that:.

ξ8g

≈ 1.2Σx1/2 (TrKK∗)2

(98)

where

g =g1

sin θwNow, if eq.(98) will hold at any scale μ for given independent values of the NCG

parameters x, Σ and TrKK∗, the corresponding β-functions βξ8 and βg have to verify thesame relation of Eq.(98). However, the one loop β-functions with complicated expressions

in terms of the various couplings (see Eq.(99)) do not seem to satisfy this relation for

any values of x, Σ and TrKK∗. Therefore, at the one loop order, the constraint Eq.(98) is not preserved under the renormalization flow. To see the complexity of the β-

functions expressions, we have derived (based on refs.[26] , [27] and [28])some of the

coupling constants renormalization group equations (R.G.) for the two doublets Higgs

model:

16π2dgidt

= big3i ; (b1,b2,b3)=( 115 ,−3,−7)

16π2dgtdt

= (Λ1+Λ2g2t )gt; Λ1=− 1720g21− 94g22−8g23 ; Λ2= 9216π2

dgbdt

= (Λ3+Λ4g2t )gb; Λ3=− 14g21− 94g22−8g23 ; Λ4= 1216π2

dξ1dt

= 24ξ21+2ξ25+ξ5ξ8+2ξ28−(9g22+ 95g21)ξ1+( 98g42+ 27200g41)+ 940g22g21+12g2t ξ1−6g4t (99)

16π2dξ2dt

= 24ξ22+2ξ25+ξ5ξ8+2ξ28−(9g22+ 95g21)ξ2+( 98g42+ 27200g41)+ 940g22g21+12g2b ξ2−6g4b16π2

dξ5dt

= 12ξ5ξ1+24ξ8ξ1+12ξ5ξ2+24ξ8ξ2− 12 (9g22+ 95g21)ξ5+( 98g42+ 27200g41)− 940g22g21+3g2t ξ5−6g2b g2t


and

16π2dξ8dt

= 32ξ28 + 4ξ5ξ8 + 2ξ25 − (9g22 +

9

5g21)ξ8 + 6(g

2t + g

2b )ξ8 − 6g2bg2t

Here g1, g2, g3, gt and gb denote the electromagnetic, weak, strong, top Yukawa and bot-

tom Yukawa couplings respectively. Moreover, an SU(5) normalization is used for which

the Weinberg θw angle is defined through the relation

g22 sin2 θw =

3

5g21 cos

2 θw (100)

and the evolution parameter t is defined as:

t = ln(μ

μ0) (101)

(μ0 is some reference scale e.g μ0 = MZ). We have also neglected all fermionic Yukawa

couplings except those of the top and bottom quarks.

We notice that if the NCG parameters x, Σ and TrKK∗ are scale dependent, theNCG relations, Schwartz and unitarity bounds (Eqs.(75),(81) and (97)) cannot be simul-

taneously verified. To be more explicit, let us assume that the relation in Eq.(73) holds

for any scale μ.Plugging the latter into the NCG relations Eq.(75), using the fact that:

TrKK∗ =4

9xg2t(102)

and the solutions of the renormalization group equations of g1, g2, gt and ξ1 as are given

by Eqs.(99), one can fix exactly the μ-dependence of the Σ parameter. However, the

latter will not be compatible with the unitarity constraint(Eq.(97)), Schwartz inequality

and the other NCG relations (Eqs.(81) and Eqs.(75)) under the renormalization flow

of the other couplings ξ2, ξ5..etc. Thus, the NCG constraints are not renormalization

group invariant and we cannot have one scale μ where all the relations are satisfied

simultaneously.

Conclusion

As a conclusion, we cannot satisfy simultaneously all the tree level NCG mass, couplings

and unitarity relations at the same energy scale and therfore we will face a sort of in-

compatibility with the renormalization group approach if the NCG constraints holds.

Thus, the two doublets higgs model can be nicely constructed classically within this ex-

tended Chamesseddine-Frohlich approach to NCG but we have to look for an appropriate

quantization procedure in the context of Noncommutative geometry.


Acknowledgement

One of us (NM)is very grateful to Profs.P.Aurenche and F.Boudjemaa for the hospitality

which I have during my stay at the LAPTH of Annecy where part of this work was

completed.We are also very indebted to Prof.J.Mimouni for useful discussions.This work

is supported by the research grant N◦D2005/13/04

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Documents

Noncommutative Geometry constraints and the Standard ...present the renormalization group (RG) analysis and show that the corresponding evo-lution diﬀerential equations, once solved