Unified Electroweak Model Based on Space-Time and Isospin Symmetries Dominique Spehler and Denis Ensminger Universit´e Louis Pasteur, I.U.T. All´ee d’Ath`enes, 67300 Schiltigheim, FRANCE and Instituto de F´ısica, Universidade de S˜ ao Paulo C.P. 66318, 05315-970 S˜ ao Paulo, SP, BRAZIL (Dated: November 22, 2005) We propose a spinorial approach to the unified electroweak interactions, in which no use is made of spontaneous symmetry breakdown. No scalar particles are needed in order to break the symmetry and no Higgs particle is left. No reference is made to gauge symmetry. Our approach stresses the role of space-time and isospin symmetries in the build up of the electroweak model. Internal degrees of freedom, such as isospin, are incorporated in the theory by using spinors carrying isospin indices. All vector bosons are described by a rank 2 field in the spinorial and the isospinorial indices. Leptons are accomodated in a rank 1 spinor field and in a rank 2 isospin field as well. The dynamical variables of the theory are the chiral and isochiral components of these fields.
I.
INTRODUCTION
The aim of physics is the description of a large class of phenomena, based on a few simple ideas and postulates. Physical theories give rise to intuitive representations and images which contribute to an understanding of the corresponding class of phenomena. A higher level of understanding is reached in physics whenever two different theories are unified. In the history of physics, some notable unification efforts were achieved.? ? ? ? ? ? After the discovery of the V -A form of the weak currents and of the possible description of weak interactions by means of intermediate vector bosons, the idea arose among physicists of a possible unified description of electromagnetic and weak forces. The most successful unification model of these two interactions is that developed by Glashow, Weinberg and Salam (G.W.S.).? ? ? The model assumes the SU (2) ⊗ U (1) group as the fundamental gauge group and its breakdown via the Higgs mechanism.? ? ? Since the SU (2) ⊗ U (1) gauge group is broken one might argue why one needs gauge symmetry to start with. The quest for other alternatives becomes relevant in view of the, up to now, lack of evidence for the existence of Higgs particle. We have already shown that within the spinorial approach gauge invariance can be derived if one assumes the kinetic term of the Lagrangian to be chiral invariant if one treats all chiral components as independent variables of the theory.? ? In this paper we show that theories in which a gauge symmetry is broken can be equally described. We propose an alternative approach to building the G.W.S. unified electroweak theory in which gauge symmetries are not required. In fact, all we need are space-time symmetries and isospin symmetry. In order to take into account the necessary Lorentz invariance of the theory, we work with spinors of rank 1 describing leptons of spin 12 and with spinors of rank 2 for the description of bosons of spin 1. We take into account the existence of internal degrees of freedom by using generalized spinor fields. The basic fields carry indices associated to isospin. The unification aspect becomes explicit in our approach since all leptons are described by one rank 1 generalized spinor field whereas all bosons are incorporated in one generalized (to include isospin) rank 2 spinor field. Since we have in our hands a large number of degrees of freedom we can build electroweak models containing more particles than the usual G.W.S. model. In this paper we deal with this minimum electroweak model. The dynamical variables of the theory are chiral and isochiral components of the basic fields. We consider among a large number of possible couplings of these components, those couplings that reproduce the G.W.S. model. The method is the following: we assume chiral and isochiral invariance? at the free level. These symmetries are broken when one considers bilinear and trilinear couplings of the isochiral-chiral components. These couplings are constructed in order to give rise to the same couplings obtained by the electroweak model of G.W.S.. The paper is organized as follows: In Section II, we recall some basic features of the spinorial approach. We give some relevant properties of the chiral components of a spin 1 field. These components are treated, in our approach, as independent field variables. In Section III we relate the notion of a lepton family to an internal symmetry, the isospin symmetry.? ? ? We show how to combinate the massive electron and his associated massless neutrino in an (2 × 2) isospin matrix. We define, in this section, isochiral components. The free field Lagrangians for leptons and vector bosons are exhibited in Section IV. These Lagrangians are a function of the chiral-isochiral components. In Section V we explain how to construct the interaction Lagrangians
2 and write the interaction Lagrangian between matter and spin 1 massless field. Having in mind the G.W.S. interactions we reduce the number of degrees of freedom that the model can accomodate in order to obtain the usual couplings of the electroweak model. The chiral and the isochiral components of spin 1 massless field are presented in Section VI. In Section VII we construct all the physically relevant quadratic interactions, in particular those giving rise to the masses of the vector bosons. We analyse some properties of the chiral-isochiral tensorial superposition factors in Section VIII and construct trilinear couplings in Section IX. The paper ends with conclusions in Section X.
II.
CHIRAL SYMMETRY AND CHIRAL COMPONENTS FOR BOSONS
The alternative construction of the electroweak model is based on the spinorial approach.? In this section we describe how spin 1 bosons can be described by a rank 2 spinor. One of the basic features of this approach is that when one associates a rank 2s spinor to particles of spin s, one is dealing with a field with (2)2s components. This makes possible an unifying description of fermions and bosons. What seems to be a major problem of the spinorial approach (too many degrees of freedom), seems to be its great advantage. The point is that, as will be shown here, the extra degrees of freedom represents all possible kinds of particles of spin s. We shall illustrate this by analysing the spin 1 case. Our proposal for the description of several species of spin 1 particles carrying internal degrees of freedom, is to associate to these particles a generalized rank 2 spinor field, that is ψα1 α2 , a1 a2 (x) .
(II.1)
The indices α1 , α2 are needed in order to incorporate internal degrees of freedom and a1 , a2 runs from 1 to 4. The problem with an apparently too large number of degres of freedom is clear at this point. Taking into account only the spinor indices, one has in hands 16 degrees of freedom. The first problem to be solved is to identify the independent variables of the theory. We have chosen to work with the four chiral components of the field ψ (ψRR , ψRL , ψLR , ψLL ) as the dynamical variables of the theory. They are defined as: 1 2 1 = 2 1 = 2 1 = 2
ψRR = ψRL ψLR ψLL
1 1 + γ5 ⊗ 1 + γ5 ψ 2 1 1 + γ5 ⊗ 1 − γ5 ψ 2 1 1 − γ5 ⊗ 1 + γ5 ψ 2 1 1 − γ5 ⊗ 1 − γ5 ψ . 2
(II.2)
Any of these chiral components transforms under the Lorentz group just like the field ψ. Furthermore, it is easy to check that ψ = ψRR + ψRL + ψLR + ψLL . (II.3) Notice that the gerators 12 1 + γ and 12 1 − γ 5 act as projection operators for the rank 2 spinor ψa1 a2 (x). Many aspects of the relevance of the spinorial approach in describing particles of spin higher than 21 ? can be understood by expanding the rank 2 spinor in terms of 16 linearly independent matrices 5
γ µ C , σ µν C , C , γ 5 C , γ 5 γ µ C
(II.4)
where C is the charge conjugation matrix. For all the matrices we adopt the properties of ref. ? . The choice of the 16 independent matrices takes into account the need for finding 10 symmetric matrices plus 6 antisymmetric matrices. The decomposition of the rank 2 spinor field in terms of the linearly independent matrices is ψa1 a2 (x) = Aµ (x) (γ µ C)a1 a2 + Fµν (x) (σ µν C)a1 a2 + AC (x)(C)a1 a2 + + A5 (x) γ 5 C a a + A5µ (x) γ 5 γ µ C a1 a2 . 1 2
(II.5)
The coefficients appearing in the expansion (??) have the following properties as far as the P, T and C, discrete symmetry transformations? ? are concerned.
3 Under space reflection, we impose that ψ transforms as: P
ψ (~x, t) −→ ψ 0 (x0 ) = ψ 0 (−~x, t) = ξP γ 0 ⊗ γ 0 ψ (~x, t) . This transformation implies that the fields Aµ , Fµν , AC , A5 and A5µ transforms under space reflection in the following way P
Aµ (x) −→ A0µ (x0 ) = − ξP (2A0 (−~x, t) g0µ − Aµ (−~x, t)) P
0 Fµν (x) −→ Fµν (x0 ) = − ξP (2Fν0 (−~x, t) g0µ − 2Fµ0 (−~x, t) g0ν + Fµν (−~x, t)) P
A5µ (x) −→ A05µ (x0 ) = ξP (2A5 0 (−~x, t) g0µ − A5µ (−~x, t)) AC (x) −→ A0C (x0 ) = AC (−~x, t) A5 (x) −→ A05 (x0 ) = − A5 (−~x, t) .
(II.6)
We also impose that the spinor ψ transforms under time reversal as T
ψ (~x, t) −→ ψ 0 (x0 ) = ψ 0 (~x, −t) = ξT γ 5 C ⊗ γ 5 C ψ ∗ (~x, −t) . This implies the following transformation properties for the superposition fields T Aµ (x) −→ A0µ (x0 ) = ξT 2 A∗0 (~x, −t) g0µ − A∗µ (~x, −t) T 0 ∗ ∗ ∗ Fµν (x) −→ Fµν (x0 ) = − ξT 2 F0µ (~x, −t) g0ν − 2 F0ν (~x, −t) g0µ + Fµν (~x, −t) T A5µ (x) −→ A05µ (x0 ) = − ξT 2 A∗5 0 (~x, −t) gµ0 − A∗5µ (~x, −t) AC (x) −→ A0C (x0 ) = − A∗C (~x, −t) A5 (x) −→ A05 (x0 ) = A∗5 (−~x, −t) .
(II.7)
Charge conjugation transformation is imposed by requiring that the rank 2 spinor transforms as: C
ψ (~x, t) −→ ψc (~x, t) = ξc γ 0 C ⊗ γ 0 C ψ ∗ (~x, t) . It follows that under charge conjugation the following transformation holds true C
Aµ (x) −→ Aµ,c (x) = ξc A∗µ (x) C
∗ Fµν (x) −→ Fµν,c (x) = ξc Fµν (x) C
A5µ (x) −→ A5µ,c (x) = ξc A∗5µ (x) C
AC (x) −→ AC,c (x) = − ξc A∗C (x) C
A5 (x) −→ A5,c (x) = − ξc A∗5 (x) .
(II.8)
In the above definitions, ξP , ξT and ξc stands, as usual, for arbitrary phase factors. Equations (II.6), (II.7) and (II.8) establish the different transformation properties of the fields Aµ , Fµν , A5µ , AC and A5 under P , T and C. Using the P , T and C transformation properties of the coefficients appearing in the expansion (??), we see that in this expansion one has 16 dynamical fields: one vector field (Aµ (x)), one pseudo-vector field (A5µ (x)), one antisymmetric tensor-field (Fµν (x)), one scalar field (AC (x)) and one pseudo-scalar field (A5 (x)). We shall see later that only the vector, the pseudo-vector and the tensor degrees of freedom are relevant for a consistent and complete description of the weak interactions. We shall drop, from now on, the scalar AC (x) and the pseudo-scalar A5 (x) fields. This simplifying choice avoids unwanted degrees of freedom. Finally, with the help of the definitions (??), and the expansion (??) (from now on we will omit AC (x) and A5 (x) )
4 we can write the chiral components as: µν 1 5 1+γ σ C Fµν (x) 2 a1 a2 1 1 + γ5 γµC Aµ (x) + A5µ (x) 2 a a 1 2 1 µ 5 Aµ (x) − A5µ (x) 1−γ γ C 2 a1 a2 1 1 − γ 5 σ µν C . Fµν (x) 2 a1 a2
ψRR (x) = a1 a2
ψRL (x) = a1 a2
ψLR (x) = a1 a2
ψLL (x) = a1 a2
(II.9)
The chiral components separates the tensor from the vector components of the rank 2 spinor field. Furthermore, one can see that in writing Euler-Lagrange equations for ψRR (ψLL ) we are, essentially, writing equations for F µν . The usual relation of tensor fields as derivatives of vector fields will naturally appear as a consequence of the equations of motion. In order to describe interactions of particles with internal degrees of freedom of the ψ field, we shall substitute the coefficients Aµ (x) A5µ (x)
by by
Aα1 α2 , µ (x) Aα1 α2 , 5µ (x)
Fµν (x)
by
Fα1 α2 , µν (x) .
and
As we shall make use later of the internal isospin symmetry, we shall expand the matrix elements Aα1 α2 , µ (x), Aα1 α2 , 5µ (x) and Fα1 α2 , µν (x) in terms of the four 2 × 2 generators σ of the SU (2) group as follows: 1 j j aµ σ α1 α2 2 1 Aα1 α2 , 5µ (x) = a 5µ j σ j α1 α2 2 1 Fα1 α2 , µν (x) = f µν j σ j α α 1 2 2 Aα1 α2 , µ (x) =
(II.10)
(sum over the j index implied; j = 0, 1, 2, 3). Using the expansion (??), we write for a spin 1 field: 1 1 j j aµ σ α1 α2 (γ µ C)a1 a2 + f µν j σ j α1 α2 (σ µν C)a1 a2 + 2 2 1 j j 5 µ + a 5µ σ α1 α2 γ γ C a1 a2 . 2
ψα1 α2 , a1 a2 (x) =
(II.11)
The point is that, when taking in account isospin, we have up to 8 boson fields at our disposition. It should be noticed that, and this will be analysed further, we are able to construct with our method more general electroweak theories (with more particles) than the usual one. We can accomodate in our approach up to twice the number of particles employing the usual G.W.S. model. III.
CHIRAL AND ISOCHIRAL COMPONENTS FOR LEPTONS
Let us turn now to the description of leptons and their interactions. In order to describe leptons we shall use a rank 1 spinor field: ηα1 α2 , a (x)
(III.1)
where, as before, the indices α1 , α2 will be related further to the internal degrees of freedom (those associated to the isospin),? ? ? and where a runs from 1 to 4.
5 In fact, as we did for the chiral components in Section II, one can define isochiral components by: 1 1 ηrr = 1 + σ3 ⊗ 1 + σ3 η 2 2 1 1 1 + σ3 ⊗ 1 − σ3 η ηr` = 2 2 1 1 1 − σ3 ⊗ 1 + σ3 η η`r = 2 2 1 1 1 − σ3 ⊗ 1 − σ3 η η`` = 2 2 in such a way that
(III.2)
η = ηrr + ηr` + η`r + η`` . (III.3) 1 0 In (??), σ 3 stands for the usual Pauli matrix, σ 3 = . 0 −1 1 3 The operators and 12 1 − σ 3 act as projection operators in the isospin space, in the same way that 2 1 + σ 1 1 5 and 2 1 − γ 5 acts in the spinor space. 2 1+γ The basic field ηα1 α2 , a (x) is represented by a 2 × 2 matrix spinor of rank 1, and can therefore be written as a linear combination of the Pauli σ matrices. One can write ηα1 α2 , a (x) =
3 X
σj
α1 α2
η ja (x)
(III.4)
j=0
or, detailing: ηα1 α2 , a (x) =
η 1a (x) − i η 2a (x)
η 0a (x) + η 3a (x)
!
η 1a (x) + i η 2a (x) η 0a (x) − η 3a (x) 1 1 1 = 1 + σ 3 η 0a (x) + η 3a (x) + σ + i σ 2 η 1a (x) − i η 2a (x) + 2 2 1 1 1 1 2 2 + σ − iσ η a (x) + i η a (x) + 1 − σ 3 η 0a (x) − η 3a (x) . 2 2 The isochiral components defined in (??) are then: 1 2 1 ηr`, a (x) = 2 1 η`r, a (x) = 2 1 η``, a (x) = 2 Defining, as usual, the chiral components by ηrr, a (x) =
1 + σ3
(III.6)
η 0a (x) + η 3a (x)
σ1 + i σ2
η 1a (x) − i η 2a (x)
σ1 − i σ2
η 1a (x) + i η 2a (x)
1 − σ3
η 0a (x) − η 3a (x) .
(III.5)
(III.7)
1 1 + γ5 η 2 1 ηL = 1 − γ5 η 2 we end up, considering both chiral and isochiral components, with 8 degrees of freedom for the description of leptons: 1 1 1 + σ3 1 ± γ 5 η 0 (x) + η 3 (x) ηrr, R (x) = 2 2 L 1 1 1 σ + i σ2 1 ± γ 5 η 1 (x) − i η 2 (x) ηr`, R (x) = 2 2 L 1 1 1 η`r, R (x) = σ − i σ2 1 ± γ 5 η 1 (x) + i η 2 (x) 2 2 L 1 1 η``, R (x) = 1 − σ3 1 ± γ 5 η 0 (x) − η 3 (x) . (III.8) 2 2 L ηR =
6 The usual chiral components ηR,a (x) and ηL,a (x) were replaced by the spinors ηα1 α2 , Ra (x) and ηα1 α2 , La (x) . a is the usual spinor index and where α1 , α2 are the internal degrees of freedom (isospin) indices. The independent dynamical variables of the theory are the isochiral components associated to chiral components of spin 12 fields η 0 , η 1 , η 2 , η 3 . ¿From purely kinematic reasons we end up with a “family”? structure. One can accomodate up to 4 fermions in each family. The concept of family, in this approach, is then associated to the isospin symmetry. Due to the extra degrees of freedom one has a natural way to extend the usual Glashow-Weinberg-Salam model. In order to reproduce the usual description, however, we choose: 1 η 0a (x) + η 3a (x) = √ ea (x) 2 1 η 1a (x) − i η 2a (x) = √ ea (x) 2 η 1a (x) + i η 2a (x) = νR,a (x) η 0a (x) − η 3a (x) = νL,a (x)
(III.9)
where ea (x) is the electron field, νL,a (x) (νR,a (x)) is its associated left-handed (right-handed) neutrino field. A priori it will be possible to introduce a massive neutrino in our theory, but in analogy to what is done in the G.W.S. model we assume that there is no right handed neutrino, and therefore take: η 1a (x) + i η 2a (x) = νR,a (x) = 0 . ¿From (III.9) and (III.10) it follows that the ηα1 α2 , a (x) matrix field associated to leptons is: 1 1 √ ea (x) √ ea (x) 2 2 ηα1 α2 , a (x) = 1 5 0 1 − γ νa (x) 2
(III.10)
(III.11)
where ea (x) and νa (x) represent respectively, the electron field and its electronic neutrino field. The decomposition (III.6) becomes now: 1 1 + σ3 α α 1 2 2 1 + 1 − σ 3 α1 α2 2
ηα1 α2 , a1 (x) =
1 1 1 1 √ ea1 (x) + σ + i σ 2 α α √ ea1 (x) + 1 2 2 2 2 1 5 1 − γ a1 a2 νa2 (x) . 2
(III.12)
Defining, as usual, η by η = η+ γ 0
(III.13)
where the plus sign acts upon both chiral and isochiral components, we obtain, for ηα1 α2 , a (x) given by the expression (??), the following expression for the 2 × 2 , η matrix: 1 √ ea1 (x) 0 2 = η α1 α2 , a1 (x) = (III.14) 1 1 5 √ ea1 (x) ν a2 (x) 1 + γ a2 a1 2 2 1 1 1 1 1 = √ ea1 (x) 1 + σ 3 α1 α2 + √ ea1 (x) σ − i σ 2 α1 α2 + 2 2 2 2 1 1 5 3 + ν a2 (x) 1 + γ a2 a1 1 − σ α1 α2 . (III.15) 2 2 Expressions (??) and (??) leads us to the following traces: Tr {η η} = η α1 α2 , a (x) ηα2 α1 , a (x) = ea (x) ea (x)
(III.16)
and Tr {η i6 ∂ η} = η α1 α2 , a1 (x) (i6 ∂)a1 a2 ηα2 α1 , a2 (x) = ea1 (x) (i6 ∂)a1 a2 ea2 (x) + 1 + ν a1 (x) 1 + γ 5 i6 ∂ νa2 (x) . 2 a1 a2
(III.17)
7 It is easy to generalize the notion of isochiral components to a spin 1 particle, described by the spinor field ψα1 α2 , a1 a2 (x) . The indices α1 α2 are again related to isospin. We define the following four isochiral quantities ψ.., RR ψ.., RL ψ.., LR ψ.., LL
= = = =
ψrr, RR + ψr`, RR + ψ`r, RR + ψ``, RR ψrr, RL + ψr`, RL + ψ`r, RL + ψ``, RL ψrr, LR + ψr`, LR + ψ`r, LR + ψ``, LR ψrr, LL + ψr`, LL + ψ`r, LL + ψ``, LL .
(III.18)
Explicit expressions for each of these chiral-isochiral components will be presented later.
IV.
FREE FIELD LAGRANGIANS
When dealing with the spinorial approach, we do not impose gauge invariance in order to build the Lagrangian. The free field Lagrangian is built by using another invariance principle: we shall impose that the free field Lagrangian is invariant under chiral transformations. For spin 21 particles, chiral transformation is defined as the following transformation of the field: 5 η → η chiral = eiθγ η = cos θ + i γ 5 sin θ η .
(IV.1)
The usual first order derivative Lagrangian L = η i6 ∂ η
(IV.2)
is compatible with chiral invariance. In terms of the right and left components, this Lagrangian is written as L = η R i6 ∂ ηR + η L i6 ∂ ηL .
(IV.3)
By using (III.18) we get for the kinetic piece of the free field Lagrangian L = e (i6 ∂) e + ν
1 1 + γ 5 i6 ∂ ν . 2
(IV.4)
The natural extension of (IV.1) to spin 1 particles, will be the following general chiral transformation: 5
5
ψ → ψ chiral = eiθ1 γ ⊗ eiθ2 γ ψ .
(IV.5)
If, in order to construct the free field Lagrangian, one imposes Lorentz invariance and requires that the Lagrangian be first order, we will get the following Lagrangian density: L0 = β1 ψ(i6 ∂ ⊗ 1)ψ + β2 ψ(1 ⊗ i6 ∂)ψ .
(IV.6)
Now, in order to get chiral invariant Lagrangian, one has just one of the two alternatives either β1 = 0 =⇒ θ1 = 0 or β2 = 0 =⇒ θ2 = 0 . Any of the alternatives lead to the same result. We shall take the second case. The Lagrangian is invariant under the chiral transformation 5
ψ → ψ chiral = eiθ1 γ ⊗ 1 ψ .
(IV.7)
It is easy to show that the free Lagrangian L0 can be written, in terms of the chiral components of ψ , as: L0 = ψ RR (i6 ∂ ⊗ 1)ψRL + ψ RL (i6 ∂ ⊗ 1)ψLL + ψ LR (i6 ∂ ⊗ 1)ψRR + ψ LL (i6 ∂ ⊗ 1)ψLR .
(IV.8)
8 In (??) a sum over the internal degrees of freedom (isospin) is implied, in such a way that all terms contributing to (??) are: ψ rr, RR (i6 ∂ ⊗ 1) ψrr, RL + ψ r`, RR (i6 ∂ ⊗ 1) ψ`r, RL + + ψ `r, RR (i6 ∂ ⊗ 1) ψr`, RL + ψ ``, RR (i6 ∂ ⊗ 1) ψ``, RL + + ψ rr, RL (i6 ∂ ⊗ 1) ψrr, LL + ψ r`, RL (i6 ∂ ⊗ 1) ψ`r, LL + + ψ `r, RL (i6 ∂ ⊗ 1) ψr`, LL + ψ ``, RL (i6 ∂ ⊗ 1) ψ``, LL + + ψ rr, LR (i6 ∂ ⊗ 1) ψrr, RR + ψ r`, LR (i6 ∂ ⊗ 1) ψ`r, RR + + ψ `r, LR (i6 ∂ ⊗ 1) ψr`, RR + ψ ``, LR (i6 ∂ ⊗ 1) ψ``, RR + + ψ rr, LL (i6 ∂ ⊗ 1) ψrr, LR + ψ r`, LL (i6 ∂ ⊗ 1) ψ`r, LR + + ψ `r, LL (i6 ∂ ⊗ 1) ψr`, LR + ψ ``, LL (i6 ∂ ⊗ 1) ψ``, LR .
(IV.9)
The conclusion is that, by imposing chiral invariance we get the following free field Lagrangian for spin 1 bosons and spin 21 fermions L00 = η(i6 ∂)η + ψ(i6 ∂ ⊗ 1)ψ .
(IV.10)
Notice, however, that the dynamical variables are the chiral-isochiral components of the η and ψ field and not the fields η and ψ .
V.
INTERACTION LAGRANGIANS — INTERACTION WITH MATTER
Chiral symmetry holds true only for ideal objects, or equivalently for free fields. For interacting fields, we expect that this symmetry will be broken. The interaction Lagrangians Lint , will be assumed to be a function of the chiral-isochiral components of all basic fields: Lint = L ψRR...R , ψRR...RL , . . . , ψLL...L , ψRR...R , . . . , ψLL...L , . . . , ψR , ψL . (V.1) | {z } | {z } | {z } | {z } 2s
2s
2s−1
2s−1
That is, we shall assume no derivative couplings of the chiral fields at the interaction level. The requirement of renormalizability of the theory leads us to consider only Lagrangians involving quadratic and trilinear couplings of the fields. Breakdown of chiral symmetry will occur at these two levels. The only interaction Lagrangian linear in the spin 1 field ψ, is the Lagrangian describing the interaction of this field with matter (represented by η). As this Lagrangian is obviously the simplest one involving spin 1 fields, we will use it in order to show that one has to reduce the number of degrees of freedom of our theory. The most general (linear in ψ) Lagrangian, describing the interaction of matter with a spin 1 field is: Lηψ = J1 η α1 α2 ψα2 α3 , RL C −1 ηα3 α1 + J2 η α1 α2 ψα2 α3 , LR C −1 ηα3 α1 + + K1 η α1 α2 ψα2 α3 , RR C −1 ηα3 α1 + K2 η α1 α2 ψα2 α3 , LL C −1 ηα3 α1 + h.c. . (V.2) The indices αj (j = 1, 2, 3) take into account the isospin degrees of freedom, and the constants J1 , J2 , K1 and K2 are arbitrary. However, since the Lagrangian must be real, we have to impose g = J1 = J2
and
K1 = K2 .
Looking at (II.9), we can see that the terms factors of K1 and K2 will involve derivatives and therefore give rise to non renormalizability.? ? ? ? ? We must set: K 1 = K2 = 0 .
9 The most general Lagrangian obtained from (??) and taking into account the above two restrictions is 1 5 µ 1 0 3 0 3 Lηψ = 2g e a + aµ + a 5µ + a 5µ γ γ e + 2 µ 2 5 µ1 1 1 1 1 2 1 2 aµ − i aµ + a 5µ − i a 5µ γ γ 1 − γ5 ν + + √ e 2 2 2 2 5 µ 1 1 1 1 5 1 2 1 2 + √ ν 1+γ a + i aµ + a 5µ + i a5µ γ γ e + 2 µ 2 2 2 5 µ 1 1 1 5 0 3 0 3 1+γ a − aµ + a 5µ − a 5µ γ γ ν + ν 2 2 µ 2
(V.3)
where in (??) we have made use of (II.9) and (III.11). As pointed out before, we have in our hands a very large number of degrees of freedom. In order to reproduce the G.W.S. well known interactions, one has to get rid of some apparently unwanted particles. We do this by imposing first:
a 5µ 1 = a 5µ 2 = a 5µ 3 = 0 .
(V.4)
Furthermore, we shall assume that all spin 1 particles are Majorana-like, that means that all the remaining fields (aµj , a 5µ j and fµν j ) are real. We define the W and W + particles as the combinations 1 a 1 − i aµ2 2 µ 1 = a 1 + i aµ2 . 2 µ
Wµ =
(V.5a)
Wµ+
(V.5b)
Finally, for reasons that will become clear later, we introduce new fields Aµ and Zµ through the linear superposition: 1 1 sin θAµ + cos θ Zµ 2 2 1 1 = − sin θAµ − 1 + 2 sin2 θ Zµ 2 4 cos θ 1 Zµ . = − 4 cos θ
aµ0 = − aµ3 a 5µ 0
(V.6)
That is, the physically relevant degrees of freedom will be the fields Aµ and Zµ which are combinations of the original fields. Although the mixing angle θ here defined have no similarity with the mixing angle θW of the weak interactions, we shall see that, from the point of view of the electroweak Lagrangian, they are the same. With the help of (??), (??) and (??) we get the following expression for Lηψ : Lηψ = L(e, ν, W ) + L(e, A) + L(e, ν, Z)
(V.7)
where g L(e, ν, W ) = √ e Wµ γ µ 1 − γ 5 ν + ν Wµ+ 1 + γ 5 γ µ e 2 L(e, A) = − g sin θ e γ µ e Aµ g L(e, ν, Z) = 1 − 4 sin2 θ e γ µ e Zµ − e γ 5 γ µ e Zµ + ν 1 + γ 5 γ µ Zµ ν . 4 cos θ
(V.8) (V.9) (V.10)
Equations (??), (??) and (??) permit us to identify θ with the usual mixing angle θW of the G.W.S. standard model. Wµ , Wµ+ are the charged boson fields, Aµ is the usual electromagnetic field and Zµ is the Z0 particle field.
10 VI.
CHIRAL AND ISOCHIRAL COMPONENTS OF THE SPIN 1 FIELD
Let us write the expression of the field ψα1 α2 , a1 a2 in terms of the gauge bosons Wµ , Wµ+ and Zµ and to the electromagnetic field Aµ . We obtain: 1 ψα1 α2 , a1 a2 = − sin θ Aµ 1 + σ 3 α α (γ µ C)a1 a2 1 2 2 1 1 2 + 1 1 2 σ + i σ α1 α2 + Wµ σ − iσ (γ µ C)a1 a2 + Wµ 2 2 ( 1 1 1 − 4 sin2 θ Zµ 1 + σ 3 α1 α2 (γ µ C)a1 a2 + 4 cos θ 2 1 + 3 Zµ 1 − σ 3 α α (γ µ C)a1 a2 1 2 2 ) 1 1 3 3 5 µ − Zµ 1+σ + 1−σ γ γ C a1 a2 2 2 α1 α2 +
1 f µν j σ jα1 α2 (σ µν C)a1 a2 . 2
(VI.1)
In (??), the indices a1 a2 are spinor indices and run from 1 to 4; α1 α2 are isospinor indices and run from 1 to 2. A sum over the j indices from 0 to 3 is implied. Since they are the dynamic variables of the theory it is also important to look at the chiral and isochiral components of the field ψ , defined in (??) and (??). We get 1 2 1 = 2 1 = 2 1 = 2
ψrr, RR = LL
ψr`, RR LL
ψ`r, RR LL
ψ``, RR LL
1 1 f µν 0 + f µν 3 1 ± γ 5 σ µν C 2 2 1 1 σ1 + i σ2 f 1µν − if 2µν 1 ± γ 5 σ µν C 2 2 1 1 1 σ1 − i σ2 f µν + if 2µν 1 ± γ 5 σ µν C 2 2 1 1 1 − σ3 f 0µν − f 3µν 1 ± γ 5 σ µν C 2 2 1 + σ3
(VI.2)
and sin2 θ 1 Zµ 1 + γ5 γµC cos θ 2 1 σ 1 + i σ 2 Wµ 1 + γ5 γµC 2 1 σ 1 − i σ 2 Wµ+ 1 + γ5 γµC 2 1 1 1 − σ3 Zµ 1 + γ5 γµC 2 cos θ 2
1 2 1 = 2 1 = 2 1 = 2
1 + σ3
ψrr, RL = ψr`, RL ψ`r, RL ψ``, RL
− sin θAµ −
(VI.3)
and, finally ψrr, LR = ψr`, LR = ψ`r, LR = ψ``, LR =
1 2 1 2 1 2 1 2
1 1+σ − sin θAµ + 1 − 2 sin2 θ Zµ 2 cos θ 1 σ 1 + i σ 2 Wµ 1 − γ5 γµC 2 1 σ 1 − i σ 2 Wµ+ 1 − γ5 γµC 2 1 1 1 − σ3 Zµ 1 − γ5 γµC . cos θ 2 3
1 1 − γ5 γµC 2
(VI.4)
As a natural generalization of rank 1 spinors, we define the field variable ψ as ψ = ψ+ γ 0 ⊗ γ 0
(VI.5)
11 or more explicitly ψ α1 α2 , a1 a2 = ψ + α1 α2 , a3 a4 γ 0 a3 a1 γ 0 a4 a2 .
(VI.6)
The spinor indices a1 , a2 , a3 , a4 runs from 1 to 4 whereas the isospinor indices α1 , α2 runs from 1 to 2. It is straightforward to get from the definition (VI.5) the bared chiral-isochiral fields associated to ψ .
VII.
QUADRATIC INTERACTIONS
We shall construct in this section all possible quadratic interaction terms. They are the simplest contributions and, furthermore they give rise to the masses of several particles. Let us consider first the quadratic interaction of the η field. The most general quadratic interaction term is Lm = − m η η = − m (η R ηL + η L ηR ) .
(VII.1)
Lm = − m e e
(VII.2)
Substituting (??) in (??) we obtain:
which means that m has to be identified with the electron mass. The left-handed neutrino appears naturally massless with our choice. Taking into account the quadratic term (??), we get therefore with the choice (??): L0matter = η i6 ∂ η − m η η = e (i6 ∂ − m) e + ν
1 1 + γ 5 i6 ∂ ν . 2
(VII.3)
We end up, in this way, with a massive electron and a massless left handed neutrino. The chiral and isochiral components obtained in Section VI, permit us to generate a mass for the gauge bosons Z and W and also to avoid a mass term for the photon. The most general coupling giving rise to a quadratic term in the Z field or equivalently to a mass term for the Z boson and avoing a mass term for the photon is: LZ = BZ ψ ``, RL ψ``, RL + ψ ``, LR ψ``, LR (VII.4) where BZ in (??) is an arbitrary constant. Performing the trace (??) we get LZ = −
1 5 BZ Z µ Zµ . 4 cos2 θ
(VII.5)
Therefore the mass of the Z boson will be mZ 2 = −
5 1 BZ . 4 cos2 θ
(VII.6)
On the other hand, the most general coupling giving rise to a mass term for the W bosons is LW = BW ψ r`, RL ψ`r, RL + ψ `r, RL ψr`, RL + ψ r`, LR ψ`r, LR + ψ `r, LR ψr`, LR
(VII.7)
where BW is again an arbitrary constant. BW will be related to the mass of the W boson. It is straightforward to obtain, from (??): LW = − 4 BW W +µ Wµ .
(VII.8)
mW 2 = − 4 BW .
(VII.9)
The mass of the W bosons is then
We can see, from (??) and (??) that the quocient usual values of mZ and mW for the choice
mZ 2 mW 2
is naturally proportional to
5 BZ = 16 BW .
1 cos2 θ
and that we obtain the (VII.10)
12 The chiral and isochiral asymmetry arguments permitted us to generate the masses of the gauge bosons. Furthermore these masses can be arranged to obey the same relation as that obtained in the standard model by using the Higgs mechanism. On the other hand, the masslessness of the photon is also preserved. Finally we will analyse the remaining bilinear couplings. Since we have succeeded in giving a mass to the Z boson, and left the photon massless with the help of the quadratic term (??), the only remaining nonzero terms to be considered are: L1 int = B1 ψ rr, RR ψrr, LL + ψ rr, LL ψrr, RR + ψ ``, RR ψ``, LL + ψ ``, LL ψ``, RR + + B2 ψ r`, RR ψ`r, LL + ψ r`, LL ψ`r, RR + ψ `r, RR ψr`, LL + ψ `r, LL ψr`, RR . (VII.11) Using (??), we get B1 ψ rr, RR ψrr, LL + ψ rr, LL ψrr, RR + ψ ``, RR ψ``, LL + ψ ``, LL ψ``, RR = n µν o µν = − 2 B1 f ∗ 0 + f ∗ 3 µν f 0 + f 3 + f ∗ 0 − f ∗ 3 µν f 0 − f 3 whereas, for the second piece of L1 int we get B2 ψ r`, RR ψ`r, LL + ψ r`, LL ψ`r, RR + ψ `r, RR ψr`, LL + ψ `r, LL ψr`, RR = n µν o µν . = − 2 B2 f ∗ 1 + if ∗ 2 µν f 1 − if 2 + f ∗ 1 − if ∗ 2 µν f 1 + if 2
(VII.12)
(VII.13)
It follows from (VII.11) – (VII.13) that not all quadratic terms are associated to masses of the particles. Their interpretation will be analysed in the next sections.
VIII.
PROPERTIES OF THE CHIRAL-ISOCHIRAL SUPERPOSITION TENSOR FACTORS
Before analysing the cubic interactions involving the components of ψ(x) and ψ(x) we shall summarize, in this section, the set of Lagrangians obtained by us, in dealing with the chiral and isochiral components as independent field variables. All Lagrangians obtained so far can be written as L = L00 + Lm + Lηψ + LZ + LW + L1 int
(VIII.1)
where L00 is the kinetic energy piece of the Lagrangian. In terms of the field η and ψ L00 is given by (IV.10). The quadratic interaction of the η (Lm ) field gives rise to the masses of the leptons. The expression for Lm is given by (??) and (??). The interaction of leptons with the boson fields is given by the term Lηψ . The explicit expression for Lηψ involves three terms, given by (??), and these three terms are summarized by expressions (??), (??) and (??). The mass terms of the Z 0 particle and the W ’s are, respectively, LZ and LW . Their expression definition are (??) and (??). The interaction of other chiral-isochiral components is represented by L1 int in (??). Its explicit representation is (??). Let us analyse some Euler-Lagrange equations that follow from (??). We shall consider the equations obtained by treating ψ.., RR and ψ.., LL as independent variables. By deriving L with respect to each variable the equations are: ψ rr, RR
is :
i6 ∂ ψrr, RL + B1 ψrr, LL = 0
(VIII.2)
ψ r`, RR
is :
i6 ∂ ψ`r, RL + B2 ψ`r, LL = 0
(VIII.3)
ψ `r, RR
is :
i6 ∂ ψr`, RL + B2 ψr`, LL = 0
(VIII.4)
ψ ``, RR
is :
i6 ∂ ψ``, RL + B1 ψ``, LL = 0
(VIII.5)
ψ rr, LL
is :
i6 ∂ ψrr, LR + B1 ψrr, RR = 0
(VIII.6)
ψ r`, LL
is :
i6 ∂ ψ`r, LR + B2 ψ`r, RR = 0
(VIII.7)
ψ `r, LL
is :
i6 ∂ ψr`, LR + B2 ψr`, RR = 0
(VIII.8)
ψ ``, LL
is :
i6 ∂ ψ``, LR + B1 ψ``, RR = 0 .
(VIII.9)
13 The substitution of the chiral-isochiral components by their expressions (??), (??), (??), and without taking into account the cubic terms we end up with the following results: ( ) eµν 1 Z 1 2 − 2 sin θ Fµν + 1 − 4 sin θ Zµν + i (VIII.10) (f0 + f3 )µν = 2 B1 2 cos θ 2 cos θ 1 3 i e (f0 − f3 )µν = Zµν + Zµν (VIII.11) 2 B1 2 cos θ 2 cos θ 1 + 1 (f1 + if2 )µν = Wµν ≡ ∂µ Wν+ − ∂ν Wµ+ (VIII.12) B2 B2 1 1 Wµν = ∂µ Wν − ∂ν Wµ+ (VIII.13) (f1 − if2 )µν = B2 B2 where Fµν = ∂µ Aν − ∂ν Aµ
(VIII.14)
Zµν = ∂µ Zν − ∂ν Zµ .
(VIII.15)
and
The conclusion is that some quadratic terms in the Lagrangian lead to the expressions of f0 , f3 , f2 and f1 as derivatives of the spin 1 bosons of the theory. The complete dependence of f0 + f 3 µν and (f1 ± if2 )µν on the above fields and the vector bosons taking into account trilinear couplings will be considered. IX.
TRILINEAR COUPLINGS
In this section we shall analyze the most general trilinear pure gauge fields interaction. To this end let us consider all possible cubic interactions involving ψ or ψ at most one derivative in the fields. In order to be compatible with CP T invariance, we have to consider, as far as trilinear couplings of ψ’s and ψ’s are concerned, the following alternatives: `I = ψ C −1 α1 α2 , a1 a2 ψ C −1 α2 α3 , a2 a3 ψ C −1 α3 α1 , a3 a1 (IX.1) `II = C ψ α1 α2 , a1 a2 C ψ α2 α3 , a2 a3 C ψ α3 α1 , a3 a1 (IX.2) (IX.3) `III = ψα1 α2 , a1 a2 C −1 ψ α2 α3 , a2 a3 ψ α3 α1 , a3 a1 (IX.4) `IV = ψα1 α2 , a1 a2 ψ α2 α3 , a2 a3 C ψ α3 α1 , a3 a1 . For Hermitian fields, the following identity holds true: C ψ α α , a a = − ψ C −1 α 1
2
1 2
1 α2 , a1 a2
.
(IX.5)
As a result of (??), one can write `I = − `II = − `III = `IV . Due to (??), the most general cubic interaction in the field ψ , or its components, can be written as Lψ = ψ C ψ ψ .
(IX.6)
(IX.7)
Let us begin considering the chiral components first. Assuming that the chiral-isochiral components are independent, there are eight types of nonzero couplings of the form (??). They are: h i M1 ψ α1 α2 , RR C ψ α2 α3 , LR ψα3 α1 , RL + ψ α1 α2 , LL C ψ α2 α3 , RL ψα3 α1 , LR + + M2 ψ α1 α2 , RL C ψ α2 α3 , RR ψα3 α1 , LR + ψ α1 α2 , LR C ψ α2 α3 , LL ψα3 α1 , RL + + M3 ψ α1 α2 , RL C ψ α2 α3 , LR ψα3 α1 , RR + ψ α1 α2 , LR C ψ α2 α3 , RL ψα3 α1 , LL + + M4 ψ α1 α2 , RR C ψ α2 α3 , RR ψα3 α1 , LL + ψ α1 α2 , LL C ψ α2 α3 , LL ψα3 α1 , RR . (IX.8)
14 Notice that the expressions (??) are written as isochiral symmetric expressions. As before, renormalizability? ? ? ? ? requires dropping the last term in (??). We then set M4 = 0. Now, assuming that the constants are real, and taking into account the hermiticity of the fields, one can show that the following identities holds true in the cubic self interactions of the gauge fields ψ α1 α2 , RR C ψ α2 α3 , LR ψα3 α1 , RL + ψ α1 α2 , LL C ψ α2 α3 , RL ψα3 α1 , LR = = ψ α1 α2 , RL C ψ α2 α3 , RR ψα3 α1 , LR + ψ α1 α2 , LR C ψ α2 α3 , LL ψα3 α1 , RL = = ψ α1 α2 , RL C ψ α2 α3 , LR ψα3 α1 , RR + ψ α1 α2 , LR C ψ α2 α3 , RL ψα3 α1 , LL . (IX.9) The conclusion is that the most general cubic, isochiral symmetric interaction is of the form: ψ α1 α2 , RR C ψ α2 α3 , LR ψα3 α1 , RL + ψ α1 α2 , LL C ψ α2 α3 , RL ψα3 α1 , LR
(IX.10)
and obviously its Hermitian conjugates. Once chirality has been analysed we now turn to the isochiral components. The isochiral components can be related as follows to the fields or their derivatives: ψrr, .. (resp. ψ rr, .. ) depends on the electromagnetic and neutral vector boson fields or their derivatives. ψ``, .. (resp. ψ ``, .. ) depends only on the neutral boson field and its derivatives. ψr`, .. and ψ`r, .. (resp. ψ r`, .. and ψ `r, .. ) depends on the vector boson field W and W + . We look, in what follows, to all the chiral-isochiral cubic interactions, giving rise to a non-vanishing trace in spinor and isospinor indices. Our guide are the interactions described by the G.W.S. model. We start considering all the chiral-isochiral interactions containing ψ rr, RR . That is, the general interaction under the form ψ rr, RR C ψ .. , LR ψ.. , RL (IX.11) resp. ψ rr, LL C ψ .. , LR ψ.. , RL . Since ψ rr, RR (resp. ψ rr, LL ) involves the fields F µν and Z µν , and since we wish only interactions of these two fields with the vector bosons, we exclude the term ψ rr, RR C ψ rr, LR ψrr, RL (IX.12) resp. ψ rr, LL C ψ rr, RL ψrr, LR and consider the interaction terms, defined now as Lψ1 . That is Lψ1 = ξ1 ψ rr, RR C ψ r`, LR ψ`r, RL + ψ rr, LL C ψ r`, RL ψ`r, LR . The same argument applied to ψ ``, RR (resp. ψ ``, LL ) allows us to exclude the terms ψ ``, RR C ψ ``, LR ψ``, RL resp. ψ ``, LL C ψ ``, RL ψ``, LR and consider now a second piece of the Lagrangian defined by Lψ2 : Lψ2 = ξ2 ψ ``, RR C ψ `r, LR ψr`, RL + ψ ``, LL C ψ `r, RL ψr`, LR .
(IX.13)
(IX.14)
(IX.15)
The choice ξ1 = − ξ2 gives rise to some terms of the usual interactions of the G.W.S. model. We shall consider now the interactions under the form ψ r`, RR C ψ .. , LR ψ.. , RL resp. ψ r`, LL C ψ .. , RL ψ.. , LR .
(IX.16)
(IX.17)
Here we must look at two group of cubic terms; one group containing the isospin rr component (containing both, electromagnetic field and gauge boson field Z ) and the other one containing the isospin `` component and describing the interaction of the spin 1 boson Z .
15 The most general interactions of the form (IX.17) are given by Lψ3 and Lψ4 , where Lψ3 = ξ3 ψ r`, RR C ψ `r, LR ψrr, RL + ψ r`, LL C ψ `r, RL ψrr, LR
(IX.18)
and Lψ4 = ξ4 ψ r`, RR C ψ ``, LR ψ`r, RL + ψ r`, LL C ψ ``, RL ψ`r, LR .
(IX.19)
The choice ξ4 = − ξ3 gives rise to other trilinear coupling of the G.W.S. model. Finally, since the superposition factor f 1 − if 2 µν (related to the isochiral component `r and to the W boson) must be the complex conjugate of the superposition factor f 1 + if 2 µν (which is related to the isochiral component r` ) we obtain the last trilinear interaction; therefore consider Lψ5 = ξ3 ψ `r, RR C ψ rr, LR ψr`, RL + ψ `r, LL C ψ rr, RL ψr`, LR − ψ `r, RR C ψ r`, LR ψ``, RL − ψ `r, LL C ψ r`, RL ψ``, LR . (IX.20) We are now able to write the complete Lagrangian. As this expression is quite long we write it in the following abbreviate form: L = L + Lψ1 + Lψ2 + Lψ3 + Lψ4 + Lψ5
(IX.21)
where L was described in the last section and Lψ1 , ..., ψ5 are trilinear couplings considered here. ¿From the Euler-Lagrange equations we get the following expressions for the superposition coefficients f 0 ± f 3 µν and f 1 ± if 2 µν . We get: 1 1 1 − 4 sin2 θ Zνµ + − 2 sin θ Fνµ + f 0 + f 3 µν = 2 B1 2 cos θ i eµν − 2i ξ1 Wµ Wν+ − Wν Wµ+ + Z (IX.22) 2 cos θ 1 3 i eµν + 2i ξ1 Wµ+ Wν − Wµ Wν+ Zνµ + Z (IX.23) f 0 − f 3 µν = 2 B1 2 cos θ 2 cos θ o 1 n + Wνµ + i ξ3 (sin θAα − cos θ Z α ) gαν Wµ+ − gµα Wν+ f 1 + if 2 µν = (IX.24) B2 n o 1 f 1 − if 2 µν = Wνµ + i ξ3 (sin θAα − cos θ Z α ) (gαµ Wν − gαν Wµ ) . (IX.25) B2 Performing now the calculations of the trilinear couplings, we get, in view of the identity ξ1 = −ξ2 : ξ1 i (cos θZ µν − sin θF µν ) Wµ+ Wν − Wµ Wν+ B1 2 ξ1 +4 W + µ Wµ+ (W ν Wν ) − W + µ Wµ . B1
Lψ1 + Lψ2 = −
(IX.26)
The choice ξ1 g = − B1 2
(IX.27)
gives rise to the usual results. Considering now ξ3 = −ξ4 we get: ξ3 + i (cos θZ µ − sin θAµ ) Wµν W + ν − Wµν Wν B2 ξ3 −2 (cos θZ µ − sin θAµ ) (cos θZµ − sin θAµ ) W + ν Wν B2 − (cos θZ ν − sin θAν ) Wν+ Wµ .
Lψ3 + Lψ4 = − 2
(IX.28)
16 The choice ξ3 g = − B2 2
(IX.29)
gives rise to the usual result. It is straightforward to verify that ∗
Lψ5 = (Lψ3 + Lψ4 )
(IX.30)
Lψ1 + Lψ2 + Lψ3 + Lψ4 + Lψ5 = L(W, A, Z) + L(W, W )
(IX.31)
in such a way that
where L(W, A, Z) is the standard interaction between W bosons and the electromagnetic and Z fields and L(W, W ) is the self-interaction of the W bosons.? ? ? X.
CONCLUSIONS
In this paper we developed an approach to the unified weak interactions in which no use is made of gauge symmetry neither of the spontaneous breakdown of these symmetries. The basic input of our purposal for the model building is that the description of weak and electromagnetic interactions requires spinor fields carrying isospin indices. That is, all fundamental fields carries spinor indices as well as indices associated to internal degrees of freedom. This approach stresses the role of internal and space-time symmetries as fundamental symmetries of nature. We considered first the case of leptons. In the description of leptons we use a rank 2 isospinor and a rank 1 spinor ηα1 α2 , a (x) . As explained in Section III the above field can accomodate four types of fermions. Since we have evidence for just two of them we showed how a massless left-handed neutrino and a massive electron can be described by the generalized spinor (a spinor with four components associated to isospin). It is possible to conjecture that isospin sets the stage for understanding the family structure. For other leptons one has to use another basic field (one for each family). That is η e α1 α2 , a (x) , η µ α1 α2 , a (x) , η τ α1 α2 , a (x) . For particles of spin 1 we use a rank 2 field in both the spinor and isospinor indices ψα1 α2 ,a1 a2 (x) . The rank 2 spinor field describes eight types of spin 1 boson fields. Four of them are used to build the electroweak model. The dynamical variables of the theory are the isochiral-chiral components of these two fields. The Lagrangian is built up by taking into account just two criteria. At the free field level we require that the Lagrangian be chiral invariant. At the interaction level we discard the terms that lead to a massive photon and those terms that lead to a non renormalizable theory. Since we make no use of gauge invariance and spontaneous breakdown of gauge invariance, no Higgs field is left.? ? ? ? ? If for a matter of consistency of the theory with renormalizability, we need the Higgs, we can add a scalar (Higgs) field at the very end. Except for the coupling of the Higgs field, the derived Lagrangian leads to the same phenomenology as the usual Glashow-Weinberg-Salam model.? ? ? Particularly, our prediction for the masses of the intermediate vector bosons, gV and gA are the same as the G.W.S. model. We conclude that in the description of the electroweak interaction there is no need of the SU (2) × U (1) gauge group neither spontaneous breakdown of gauge symmetry. Our model accomodates more particles than the G.W.S. model does. For leptons we can predict a family structure that can accomodate two extra leptons. As far as the bosons is concerned we can add, in a very natural way, four extra spin 1 bosons. Conceptually this approach to the electroweak interactions is different from the usual one proposed by Glashow, Weinberg and Salam. The electroweak theory emerges by using different principles. This alternative approach might become, however, the only alternative if new vector particles and leptons are observed in the future or if the Higgs is not found.
17 Acknowledgments
´ We would like to thank Profs. O.J.P. Eboli and A.J. da Silva for very estimulating discussions on this subject. This work was partially supported by Funda¸ca˜o de Amparo `a Pesquisa do Estado de S˜ao Paulo (FAPESP), Conselho Nacional de Desenvolvimento Cient´ıfico e Tecnol´ogico (CNPq) and by Programa de Apoio a N´ ucleos de Excelˆencia (PRONEX).
Pauli, W., 1921, “Relativit¨ atstheorie”, in Encyclopaedia der Mathematischen Wissenschaften (Leipzig). Weyl, H., 1918, “Raum, Zeit, Materie” (Berlin). Einstein, A., “The meaning of relativity” (Blackie). Kaluza, T., and Preuss, S.B., 1921, Akad. Wiss. Math. Natur., 966. Bergmann, P.G., 1942, “An introduction to the theory of relativity” (New York). Tonnelat, M.A., 1950, “Theories unifi´ees de la gravitation et de l’electromagnetisme” (Masson). Glashow, S.L., 1961, Nucl. Phys. 22, 579. Weinberg, S., 1967, Phys. Rev. Lett. 19, 1264. Salam, A., 1968, in “Elementary Particle Theory”, N. Svartholm, ed., Proceedings of the 8th Nobel Symposium (Almqvist and Wiksell, Stockholm). Goldstone, J., 1961, Nuovo Cimento 19, 154. Higgs, P.W., 1964, Phys. Lett. 12, 132; Phys. Rev. 145, 1156 (1966). Kibble, T.W., 1967, Phys. Rev. 155, 1554. Marques, G.C., and D. Spehler, 1996, J. Math. Phys. 37, 174; Int. J. Mod. Phys. A14, 5121 (1999); “Magnetic Monopoles and Chiral Asymmetry” (to appear). Marques, G.C., and D. Spehler, 1999, Mod. Phys. Lett. A14, 1317. Kirchhach, M., 1992, in S.C. Wittink, C.D. Goodman, G.E. Walker, eds., “Spin and isospin in nuclear interactions” (New York and London, Plenum Press) and references therein. Tuve, M.A., N. Heydenberg, and L.R. Hafstad, 1936, Phys. Rev. 50, 806. Breit, G., E.V. Condon, and R.D. Present, 1936, Phys. Rev. 50, 825. Breit, G., and E. Feenberg, 1936, Phys. Rev. 50, 850. Bargmann, V., and E.P. Wigner, Proc. Natl. Acad. Sci. 34, 211. Bjorken, J.D., and S.D. Drell, 1964, “Relativistic quantum mechanics” (New York, McGraw-Hill) and “Relativistic quantum fields” (New York, McGraw-Hill, 1965). Muirhead, H., 1968, “The physics of elementary particles” (Pergamon Press). Mohapatra, R.N., 1991, “Massive neutrinos in physics and astrophysics” (Singapore, New Jersey, World Scientific). Hagiwara, K., et al., 2002, Phys. Rev. D66. Schweber, S.S., 1961, “An introduction to relativistic quantum field theory” (New York, Harper & Row). Bogoliubov, N.N., and D.V. Shirkov, 1959, “Introduction to the theory of quantized fields” (New York, John Wiley & Sons). Abers, E.S., and B.W. Lee, 1973, Phys. Rep. 9C, 1. Taylor, J.C., 1976, “Gauge theories of weak interactions” (Cambridge University Press). Nash, C., 1978, “Relativistic quantum fields” (New York, Academic Press). Leite Lopes, J., 1983, “Gauge fields theories: An introduction” (Pergamon Press). Weinberg, S., 1996, “The quantum theory of fields” (Cambridge University Press). Appelqvist, T., and C. Bernard, 1980, Phys. Rev. D22, 200. Longhitano, A.C., 1980, Phys. Rev. D22, 1166. Appelqvist, T., 1980, in K.C. Brower and D.G. Sutherland, eds.,“Gauge Theories and Experiments at High Energies” (Scottish University Summer School in Physics, St. Andrews). Longhitano, A.C., 1981, Nucl. Phys. B188, 118. The LEP Working Group for Higgs Bosons Searches, preprint LHWG — Note/2002/01.
