Centre de Physique Théorique - CNRS - Luminy, Case 907 F-13288 Marseille Cedex 9 - France
We make a short review of the formalism that describes Higgs and Yang Mills fields as two particular cases of an appropriate generalization of the notion of connection. We also comment about the several variants of this formalism, their interest, the relations with noncommutative geometry, the existence (or lack of existence) of phenomenological predictions, the relation with Lie super-algebras etc.
anonymous ftp or gopher: cpt.univ-mrs.fr Keywords: Higgs, standard model, electroweak interactions, non commutative geometry
Work supported in part by a PROCOPE project between Mainz University and CPT Marseille-Luminy.
Based in part on lectures given by R.C. at
the VI Simposio Argentino de Física Teórica de Particulas y
Campos, Bariloche (Argentina, January 95) and at the Schladming
Winter School (Austria, March 95)
This paper is written for those who are not willing to become experts in the field of noncommutative geometry but nevertheless want to understand the link between this approach and the usual formulation of the Standard Model of electro-weak interactions. The paper tries to give simple answers to the following questions:
The construction of
the full standard model (with usual quarks and leptons but also with
right neutrinos) is carried out by following the simplest possible
route (at least the simplest, from the point of view of the present
authors) and using an appropriate generalization of the notion of
connection. The present paper can be considered as a sequel of
 but can also be read independently; it should not be
considered as an expository lecture on the approach initiated by
2. The meaning of noncommutative geometry
From the point of view of Physics, one can summarize the situation very simply by saying that ``commutative geometry'' is the collection of mathematical tools describing classical physics whereas ``non commutative geometry'' is the collection of mathematical tools describing quantum physics.
Commutative geometry (or better ``commutative mathematics'') deals with mathematical properties of spaces (measurable, topological, differentiable, riemannian, homogeneous...). For the physicist, these "spaces" provide a mathematical model for the system under study and all the properties of interest can be expressed in terms of an appropriate class of (numerical) functions defined on such spaces. It is a fact - physically obvious but also mathematically rooted - that properties of "spaces" are entirely encoded in terms of properties of algebras of numerical functions (coordinates for example) or of objects themselves defined from numerical functions (forms, tensors etc.) The name ``commutative mathematics'' comes from the fact that a set of numerical functions defined on a space is a commutative (and associative) algebra for pointwise multiplication and addition of functions.
Non commutative geometry (or better ``non commutative mathematics'') deals with mathematical properties of algebras which are not necessarily commutative and generalizes - or tries to generalize - the constructions already known for commutative algebras (i.e. spaces) to non commutative situations (i.e. to operators).
This shows that one should maybe not always speak of ``commutative geometry'' or of ``non commutative geometry'' but of ``commutative mathematics'' or of ``non commutative mathematics''. What we have in mind in the present paper is not the use of non commutative mathematics in physics (because this could include, among many other things, the mathematics of quantum statistical mechanics) but non commutative differential geometry.
epistemological point of view, and once the concepts of commutative
geometry and/or non commutative geometry have been mathematically
studied, one should probably revert the first general statement of
the present paragraph and define classical physics itself as a human
activity characterized by the wish of understanding what we call
"Nature" in terms of commutative mathematics and define quantum
physics in the same way but where the models are now expressed in
terms of non commutative mathematics. One could even go further and
declare that only the choice of the mathematical model (or models)
gives a meaning (meanings) to the whole thing (Nature) and that there
is no such thing as ``reality'' per se... but we now abandon these
philosophical considerations to return to the differential calculus,
commutative or not.
3. Non commutative versus commutative differential
A branch of non commutative differential geometry is non commutative
differential calculus. The aim is to be able to consider objects like
df or , i.e. differentials or covariant differentials,
and to perform computations with them, assuming that f is no longer
a function but an operator acting in some Hilbert space. Such a
calculus has been developed in the recent years. There exist several
kinds of non commutative differential calculi (for instance
[2, 3, 4, 5, 9, 10]) and we do not intend here to
describe them all. As a matter of fact, we shall describe none of
them. Indeed, it turns out that a very simple by-product of this
(these) generalization(s) gives us the necessary tools to understand
Higgs fields as generalization of connections (Yang Mills fields). In
some cases, for instance the model studied in
, this by-product actually belongs to the realm of
commutative geometry because it involves only commutative algebras of
functions on "spaces"! The point is that it was discovered
historically  only after the new developpements of non
commutative calculus. It is maybe a little bit misleading to call
"non commutative" some of these considerations, first because, in
simple cases, they are not so, and next because they give the reader
the feeling that he should first master all (or most of) the
niceties of non commutative differential calculus to understand the
constructions. Of course, this is a matter of taste and some people
could as well argue that one should always understand the general
before going to the particular...
4. Commutative non local differential calculus
In the previous paragraph, we said that the constructions that are at the root of our understanding of Higgs field as generalized connections do not really belong to the realm of non commutative differential geometry (they are a by-product). They however correspond to some commutative - but non local - geometry. Let us see why.
Consider a discrete set with two elements that we call L and R. Call x the coordinate function and y the coordinate function . Notice that xy=yx=0, and x + y = 1 where 1 is the unit function 1(L)=1, 1(R)=1. An arbitrary element of the associative (and commutative) algebra generated by x and y can be written (where and are two complex numbers) and can be represented as a diagonal matrix . One can write and is isomorphic with . We now introduce a differential satisfying , and the usual Leibniz rule, along with formal symbols and . It is clear that , the space of differentials of degree 1 is generated by the two independent quantities and . Indeed, the relation x+y=1 implies , the relations and imply , therefore and . This implies also, for example , , , etcMore generally, let us call , the space of differentials of degree p; the above relations imply that a base of this vector space is given by . Call and . This space is an algebra: We can multiply forms freely but one of course has to take into account the Leibniz rule, for instance . Since each is two dimensional we can easily represent it in terms of matrices. More precisely, we can represent the element as the diagonal matrix and the the element as the off diagonal matrix In other words we represent even forms by even (i.e. diagonal) matrices and odd forms by odd (i.e. off diagonal) matrices; doing so is not only natural but compulsory if we want the multiplication of matrices to be compatible with the multiplication in . Indeed, the relations
imply that the above representation using matrices is indeed a homomorphism of graded algebras from the algebra of universal forms (graded by the parity of p) to the algebra of complex matrices (with grading associated with the decomposition of a matrix into diagonal and non diagonal components). The presence of a factor i in the off diagonal matrices representing odd elements (see above expressions) is necessary for the matrix product to be compatible with the product in . Notice that the algebra is infinite dimensional (since p ranges from 0 to infinity) but if we represent the whole of in terms of matrices acting on a fixed 2-dimensional vector space, the p grading is lost and only the grading is left. The differential obeys the usual Leibniz rule when it acts on elements of but a graded Leibniz rule when it acts on elements of , namely where denotes 0 or 1 depending if is even or odd.
A one-form (this will be interpreted as a Higgs field and can be seen to define a generalized connection) is an element of . Take . The matrix representation of A reads therefore
The corresponding curvature is then , but and , so that the curvature can also be written
We now chose a hermitian product on by declaring the base to be orthonormal. Then . One recognizes here a (shifted) Higgs potential. The previous calculation (expressed in the language of K-cycles) is already discussed in [1, 2] and can be recognized in  where it is written in the language of matrices.
The previous construction could of course be generalized. For instance, we could take three points rather than two. It is easy to show that in such a case, is of dimension 6 and of dimension 12. If we take q points the dimension of is . More generally, if we take infinitely many points - take for instance points belonging to a manifold X - it is easy to see that elements of can be defined as functions A(x,y) of two variables on X such that A(x,x)=0 and that elements of can be defined as functions F(x,y,z) of three variables on X such that F(x,y,y) = F(x,x,y) = 0.
In the case of the geometry on the discrete set - that is our main example in the present paper - we recover the fact that an element A of considered as a function of two variables should satisfy the constraints A(L,L)=A(R,R)= 0 and can therefore be written as off-diagonal matrices indexed by L and R. An element F of considered as a function of three variables should satisfy the constraints F(L,L,R)=F(R,R,L)=F(L,R,R)=F(R,L,L)=F(R,R,R)=F(L,L,L)=0 so that non zero components are F(L,R,L) and F(R,L,R). The fact that for all p explains that we can use a representation of fixed dimension (namely matrices) for all values of p but one should maybe remember that it would not be so if we were considering a geometry on more than 2 points.
Notice that we are here doing commutative differential calculus
(because the associative algebra of functions on a set of 2
elements is just the commutative algebra of diagonal 2 by 2
matrices with real or complex entries) but that we are doing a non
local differential calculus because the distance between the two
points labelled L and R can not be made infinitesimally small.
The reader will have recognized that one can interpret the above
results in terms of Higgs fields. This is the subject of our next
5. What are the Higgses ? The Yukawa interaction term
Higgs fields ( ) allow left and right fermions ( ) to communicate. In four dimensional Minkowski space, this is clear from the trilinear Yukawa couplings such as that appear in the Lagrangian density of the Standard Model. This should be contrasted with terms like or where or denote usual Yang Mills gauge fields. If we had no Higgs fields, of course we would have no mass term but also no possible communication (interaction) between right and left. There would be no justification for choosing a single connected manifold to modelize our universe. We would have a Minkowski space-time for the right movers and a Minkowski space-time for the left movers. Existence of chirality in four dimensions leads therefore to the conclusion that we live in two parallel universes, one labelled by L and the other by R. Usual connections - Yang Mills fields - connect (infinitesimally) L and L together and R and R together whereas Higgs fields are non local connections that connect L and R and allow us to identify the two copies of our universe.
As explained in all books of particle physics, the scalar interaction (Yukawa) of quarks is a priori of the form
where all quark fields of charge 2/3 are collected into the multi-spinor field and similarly for quarks of charge -1/3 with The complex matrices and encode all the dimensionless Yukawa coupling constants (here spinor and scalar fields have their usual dimensions, namely 3/2 and 1 in units of mass). If we expand the previous expression, we find
Let us now collect all left-handed quark fields (of charge 2/3 and -1/3) of the standard model into a single spinor and all right-handed quark fields into a single spinor . Here and . The above Yukawa interaction term reads
The mass term is obtained by shifting and by a real constant with dimension of a mass that we call . The mass term itself is therefore described by the mass matrix
Writing , the whole fermionic lagrangian, for quarks, reads with
Where and collectively refer to those components of the gauge fields coupled to the left and right handed sectors and collectively refers as before to Higgs fields couplings. The scalar interaction (Yukawa) of leptons is exactly of the same type. The only possible difference is that, in the minimal Standard Model, one does not usually add right neutrinos. We shall actually add such right neutrinos: They will not be coupled to the gauge fields, of course, but they will give a mass to the different kinds of Dirac neutrinos and will be also coupled between themselves - via mass matrices - and to the Higgses (and also therefore, in the unitary gauge, to the longitudinal part of the gauge bosons). Notice that we do not consider Majorana neutrinos. Introducing right neutrinos, not only allows us to use the same formalism for quarks and leptons (the only difference in the Yukawa interaction term is the replacement of matrices and by and respectively) but also, as we shall see later, simplifies our analysis. The Yukawa interaction for leptons is
where all leptons fields of charge -1 are collected into the multi-spinor field and similarly for the neutrinos The complex matrices and encode all the Yukawa coupling constants. The whole fermionic lagrangian, for leptons, reads as before, but with
In the standard model, one should consider simultaneously not only the three generations of leptons but also three copies (for color) of the three generations of quarks. Taking into account - as above - the presence of three kinds of right neutrinos, we get an interaction term , with and where both and are multi-spinor fields - they are column vectors with 24 components (since 24 = 3 + 3 + 3 (3 + 3)), each component being itself a Weyl fermion.
In the spirit of
noncommutative geometry, one should think of as a
generalization of the Dirac operator (it incorporates masses and
Yukawa couplings) coupled to an algebraic connection. It should be
called the Dirac - Yukawa operator. The first piece in this
expression is a generalized differential operator since the mass
matrix appears as the inverse of a quantity encoding a
discrete set of fundamental lengths. The second piece is a
generalized connection: it incorporates both Yang-Mills and Higgs
6. The bosonic lagrangian
The theory of - usual - connections explains why is the natural object (curvature) associated with a Yang-Mills field. The root of the explanation being that the square of the corresponding covariant differential is a linear object whose expression is precisely given by the above formula. In the same way, and as explained (section 4) in a very simple case, the theory of generalized connections shows that is the natural object (curvature) associated with the Higgs field introduced in the section 4 and defined as a non local connection on the discrete set .
Now, we do not have a discrete set but a space that is the union of space-time for left-handed movers and space-time for right-handed movers, in other words, we have the product of Minkowski space by a discrete set of two elements called L and R. The generalized curvature associated with the generalized connection introduced in the previous paragraph is
The symbols and denote the usual curvatures of Yang-Mills fields associated with hermitian fields L and R. The expression of matrix elements of given before is a non trivial consequence of the formalism of non commutative geometry (or of a non local commutative differential calculus!) and can here be taken as a definition. These expressions can indeed be computed from the theory of general connections (commutative or not). The components of the curvature were obtained first by . Up to different normalization factors and the presence of spurious fields, their expression agrees with the one given just above. This analysis was later improved in  (replacement of the so-called algebra by ). A detailed exposition of the formalism of  using K-cycles and Dixmier trace can now be found in several places [3, 17, 18]. The matrix elements of given above were obtained by [4, 7] in a simple way (and using the above notations). Our method is briefly recalled in one of the ``comments'' of section 8.
Notice that the above expressions for have a dimension of a mass squared and that, as a consequence, an arbitrary mass scale appears in the formula. Explicitely, the term and its adjoint can be computed from the expressions of given previously, both in the quark and leptonic sectors.
Up to a normalization factors (we shall come back later to this physically important problem) one recognizes that the trace of is nothing else than the lagrangian describing the bosonic sector of the standard model: One obtains directly the expression that usually comes after a shift by in the Higgs fields and (see  for a discussion of this point).
In a sense, the discussion could stop at this point. Indeed, we have seen in section 5 how to re-write the Dirac-Yukawa interaction term of fermions and in this section how to recover the whole bosonic sector of the Standard Model by treating Yang Mills fields together with Higgs fields as different components of a generalized connection. However, there are several claims made in the literature about possible constraints on the parameters of the lagrangian that one could obtain thanks to a formalism of non commutative geometry. Because we want to clarify this point (at least in the present formalism) we shall continue the discussion a little further.
The whole discussion comes actually from our understanding of the notation that should denote a real number. From the one hand, if we decide to introduce, by hand, as many arbitrary constants in the expansion of this quantity (that gives rise to the full bosonic lagrangian of the standard model) as gauge invariance allows, we recover exactly the standard model with the same (unpredictive) relations as usual, namely , and where g, , and are undetermined. If, on the other hand, we decide to introduce a constant in front of in order to normalize simultaneously all the gauge fields and Higgs fields, we obtain non trivial relations. The interest of the formalism of non commutative differential geometry is not, for us, tied up with the existence of such relations; it may be, however, that such relations turn out to acquire, some day, a better status. For this reason, and also because the reader may be interested, we shall devote the end of this section to discuss them.
After global multiplication by , we can rescale gauge fields as usual by and also the Higgs fields by . Under identification with the usual lagrangian one obtains immediately ; this relation is quite natural from a point of view that identifies gauge fields and Yang Mills fields as different components of a generalized connection. In that case, the first general relation giving is not modified but the second relation becomes . Moreover, as we shall see below, the value of also gets constrained.
Rather than writing again in full the well known bosonic lagrangian of the Standard Model, we shall examine several of the terms, as they appear here. First of all, notice that one can identify the two sides of
provided . The mass value for the Higgs particle coming from this usual expression is . Notice also that the left hand side contains no additive constant (absence of cosmological term).
In our case, the Higgs potential itself coming from reads,
If we now express in terms of the component Higgs fields and in terms of the matrices of Yukawa coupings then remove the factor , in front, by rescaling the fields, we see that contains a term equal to but the term leads to a kinetic term for equal to so that the mass of the Higgs field does not depend on the mass of fermions and stays undetermined (remember that is a free parameter). Other authors , using a different formalism find quite stringent constraints relating to the fermionic masses.
The full bosonic interaction contains also a term ; using the previous expression for implies that the field L-R becomes massive, as it should. Indeed it corresponds to the Z and W bosons. One may adopt the point of view that the present formalism dictates a particular value for the Weinberg angle; this value turns out to depend upon the fermionic content of the theory. Indeed, the gauge fields L and R consist of three copies of
Here y = 1/3 for quarks since their weak hypercharge is equal to and y = 0 for leptons since their weak hypercharge is equal to (y= - 1, y = - 1; y+1 = 0 , y-1= - 2) . We are introducing here right neutrinos that are isospin singlets and for which y = 0.
For colourless quarks alone, the normalization
would lead to x = 22 / 9 and
For leptons alone, the normalization
would lead to x = 6 and
More generally, if one uses an arbitrary representation and normalize fields L and R to 1 as above, one finds
which, in the case of three families of quarks (with color) and leptons, gives (or as it is in the unified SU(5) theory. This would be therefore the ``predicted'' value for the Weinberg angle. However, in the usual approach, and even without SU(5) unification, one would obtain exactly the same value by postulating that the gauge group is not an arbitrary group isomorphic with but a group metrically isomorphic with the subgroup of SU(5). In absence of a principle based on the ideas of group symmmetries (or a generalization of such a principle), one could then ask on which grounds one should postulate such a property. The same argument (or objection) holds here. Indeed gauge invariance alone allows for the introduction of arbitrary constants in front of the individual components of the gauge group. The conclusion is therefore that, although the value appears quite ``naturally'' in this formalism, it should not be taken as an unescapable consequence of the construction.
A last possible ``constraint'' concerns the mass of the W (or Z) particle. Indeed, from the expression of we obtain a term that gives a mass to the W and the Z. The trace itself reads
This gives the relation which is well known in the standard model. In general, we have and this becomes only a constraint (namely ) if we set to the ``natural'' value as discussed before. One could hope that such relations could hold at a scale where the previous value for is experimentally satisfied (maybe at some grand unification scale). Notice that other authors , using a different formalism (relying upon the choice of another differential algebra), obtain another type of relations. Of course, we cannot (and will not) pretend that other approaches should, or not, lead to the same ``numerical'' relations. Existence of constraints such as the above ones can anyway be criticized since gauge invariance alone allows us to multiply terms of the bosonic lagrangian by arbitrary constants; this possibility can be related to the choice of particular scalar products in the space of forms ) and there are no compelling reasons to set such constants equal to one (although it may look quite natural in this formalism).
The main conclusion of this section is that the structure of the
whole bosonic lagrangian of the Standard Model can be obtained from
the formalism of non commutative geometry. Whether or not one should
look for constraints and take them seriously is another matter. Our
opinion is that, before reaching any conclusion on this line, one
should wait till we have a full understanding of the fully quantized
field theory in terms of non commutative geometry.
7. Higgs fields and super-algebras
The space where lives is naturally graded by L and R, i.e. can be decomposed into a left and a right part. Therefore transformations that map fields to themselves fall naturally into 2 kinds: those mapping L to L (and R to R) - we call them ``even''- and those mapping L to R (and conversely) - we call them ``odd''. Mathematically speaking, the space of these transformations can be considered as an associative graded matrix algebra whose corresponding Lie super-algebra is usually denoted by GL(p|q) where p (resp. q) is the number of left Weyl (resp. right) fermions entering the Lagrangian. The usual Yang-Mills fields can be decomposed onto the even part whereas the Higgs fields can be decomposed onto the odd part of this algebra. This is a rather trivial remark since any Yang-Mills theory (and not only the Standard Model) defined on an even dimensional space-time can be analysed along the same lines. Another way to express the same idea is to say that any Yang Mills theory with p left Weyl fermions and q right Weyl fermions can be formulated in terms of representation theory of some super Lie algebra posessing a representation on a graded vector space of dimension p+q. In the case of the Standard Model (with right neutrinos), and because all the fermionic species are coupled to the same gauge and Higgs bosons, the matrix describing this interaction can be decomposed on a subset of the generators of . Since we have only 4 gauge bosons and 4 Higgs bosons, we need only to use 8 generators (4 even and 4 odd ones); in other words we only need to use (or to recognize) the Lie superalgebra . The physical representations of interest (namely leptons, quarks and possibly right neutrinos) correspond to direct sums of Sl(2|1) representations of dimension 3 = 2+1, 4 = 2+2 or 1. This fact was actually observed long ago [28, 29] and sometimes perceived as a kind of ``miracle''; for us, we consider this property as almost tautological. The emergence of Lie superalgebras could lead people to think that one should try to enlarge the formalism of gauge theory to accomodate Lie superalgebras... Such attempts have been investigated in the past and shown to lead to serious problems and have, in any case, nothing to do with the Standard Model itself and even less with the non commutative geometry presentation of the Standard Model. In order to stress this point, let us consider the following analogy: one can observe that Dirac spinors form a representation of the Clifford algebra (the Dirac algebra of -matrices); this is well known; as a consequence it is also true that the spinors with four complex components also provide a representation for the (non simple) Lie algebra generated by taking commutators of arbitrary products of matrices; this does not mean that the lagrangian of quantum electrodynamics should be invariant (globally or locally) under such transformations. The fact that an algebra (like the full algebra of matrices) is not directly related with an invariance of the lagrangian does not make it useless (the spin group and its Lie algebra can of course be expressed in terms of the 's but the Clifford algebra itself is much bigger). Not all algebras related to the mathematics of a physical model need to describe ``invariances'' or ``symmetries''; the fact that they do not does not make them useless! The same thing is also true here for the super-algebra along the representation of which one can decompose the matrices acting on the vector space spanned by the multi-component spinor fields . This useful algebra is spanned by 8 generators. The first four are matrices that, in the interaction term of the lagrangian describing interaction between fermions and gauge bosons, appear as coefficients of the Yang-Mills fields and B; they are denoted, as usual, by and Y. The last four are matrices that appear as coefficients of the Higgs fields and ; they give rise (after having added the hermitian conjugate) to the Yukawa and mass interaction term. We call them , , and . More precisely, consider the following (block) matrices:
where a , b, g, e are themselves square matrices, for example of size if we consider only quarks coming in 3 families. In this case, we decide to label the basis as follows: . Let us define , and . The electric charge is Then, provided matrices e,b,g,a satisfy the relation e b + g a = 1, one can show (it is straightforward but cumbersome) that the matrices satisfy the relations
One recognizes here the usual relations defining the Lie super algebra of SL(2|1). In the case of quarks, one furthermore impose the following constraints for the hypercharge generator: with , and These constraints are satisfied if and only if, on top of the relation , the matrices e,b,g,a satisfy also the relations , and Indeed, one finds , and . This imply in particular g a = a g and e b = b e. One can then check that matrices and are then automatically what they should be.
One may notice that the above expressions for Omega matrices describing the gauge and Yukawa couplings of the quark family define a Lie superalgebra representation which is equivalent to the sum of (three) irreducible representations (each irreducible itself splits into the direct sum of a doublet and two singlets under the branching to the Lie algebra of ).
Define now and write This expression can not be real, indeed would imply , , but the other constraints would lead to a contradiction ( ). To obtain a real expression, one has to add and . Writing gives
and we recognize the expression of given in section 5, with the identification and . Warning: The matrix defined previously in terms of the matrices is not equal to the matrix defined in section 5; in order to compare the two expressions, one has to first add the conjugated expressions and . Taking into account the constraints on blocks a,g,e and b, one obtains the relations: and These relations do not imply any ``new'' constraints on mass matrices and since g and e are themselves arbitrary. The main interest of those formulae is to provided a new parametrisation for mass matrices or matrices of Yukawa couplings. This could, in turn, suggest new phenomenological ansatz for them and may even give us more insight into the structure of fermionic mass matrices. Such an ansatz was analysed in  in the case of two families and leads to a phenomenological expression of the Cabibbo angle in terms of quark masses; another ansatz for matrices a and b was analysed later in  for the case of three families.
Remark: The quantity may be thought as the contribution to the lagrangian of a particular representation of Sl(2|1). One can think of as the contribution of the antiquark representation to the lagrangian. However this identification is a little bit tricky and may lead to possible mistakes of interpretation; indeed, is not hermitian but is not the charge conjugate representation (in any case weak interactions usualy violate charge conjugation and one should not build a lagrangian that would be C-even !). Given and as before, one can define the following ``hatted'' matrices: and It is then straightforward to check that these hatted matrices generate (thanks to the same commutation relations) matrices , and , with, for example We obtain in this way a new representation (the relation being automatically satisfied since ). With as before, we can rewrite as and as
so that itself appears as the contribution associated with the ``hatted'' representation. If one wishes to use in terms of a contribution of antiparticles, for instance as , one can do it, modulo proper care, but it may be misleading.
For leptons, the idea is the same as for the quarks and, in order to straigthen even more the analogy, we add right neutrinos to the Standard Model (they will turn out to be iso-singlets, as they should be). We shall order the basis as follows: with , and define matrices Omega as previously, in terms of new block matrices e,b,g,a. However, in the case of leptons, the constraints for the hypercharge generator are different. Indeed, with , and These constraints are satisfied if and only if, on top of the relation (which ensures that commutation relations for SL(2|1) hold), the matrices e,b,g,a satisfy also the relations and ag = 0. With these constraints, one can then check that matrices , and Y defined as before in terms of the matrices are then automatically what they should be.
One may notice that the above expressions for Omega matrices describing the gauge and Yukawa couplings of the lepton family (including right neutrinos) define a Lie superalgebra representation which is equivalent to the sum of (three) reducible indecomposable representations (each of them splits into the direct sum of a doublet, a singlet, and the trivial representation under the branching to the Lie algebra of ).
Again, we define and write This expression can not be real, and, in order to obtain a real expression, one has to add, as before, and . Writing gives
We recognize the expression of given in section 5, with the identification and but the matrices a,g,e and b are not totally arbitrary since they should here satisfy the constraints and a g = 0. These relations do not imply any constraints on mass matrices and but provided a new parametrisation for them. This parametrization in terms of a,g,e,g may, in turn, suggest new phenomenological ansatz (for instance one can see what happens if these matrices a,g,e,g have particularly simple forms). Such ansatz should then be considered as educated guesses but not as ''predictions''.
Before ending this section, we would like to notice that there exists still another interesting family of parametrizations for matrices a,g,e and b. The reader can indeed check that, if we chose arbitrary ( ) matrices and choose a,g,e and b in such a way that , , and , then, all commutation relations for matrices are still satisfied. The generators and obtained from them are also equal to what they should be. However, the obtained hypercharge generator Y is not diagonal (and not necessarily hermitian) but equal to . In other words, this describes a family of quarks-like objects which are not eigenstates of hypercharge (hence of charge). The Lie superalgebra specialist may relate this possibility to the existence of reducible indecomposable representations of SL(2|1) whith non diagonal Cartan subalgebra  (take nilpotent matrices). Relation between family mixing and existence of such representations was suggested in  but was leading to difficulties (emergence of flavour changing neutral currents in the quark sector) that could only be cured by a rather ad hoc treatment of the definition of charge conjugacy. Here, we just notice that, after having defined and as before and added the (usual) complex conjugate, one obtain a real expression and one can choose to diagonalize simultaneously and Y. The rotated quark-like objects become now hypercharge (and charge) eigenstates, but the values of their charges are not standard and deviate from their usual values by corrections encoded in matrices . This last family of parametrization leads therefore to something that deviates from the Standard Model and we shall not elaborate more on this topic.
The -graded algebra discussed in this section is
not usually mentionned in textbooks explaining the construction of
the Standard Model. However, if one decides to rewrite the lagrangian
in terms of multicomponent spinor fields
gathering all left and right fermionic species in this way, this
algebra (or better representations of it) appears naturally. It
plays a role very similar to the (Clifford) Dirac algebra itself. We
suggest to call it the ``Yukawa algebra''. Again, one should not
consider this algebra as a ``symmetry'' of the model and it is
probably better to avoid the word ``symmetry'' in this context in
order to avoid possible misunderstandings.
If we take as in section 5 and define , we obtain the curvature given at the beginning of section 6. The reader should consult  for a simple exposition of this calculation. It is not necessary to use the formalism of Lie super algebras to obtain these results but one may also very well choose to use it. The definition and calculation of and in the Connes' formalism is given in [2, 3, 17].
One recognizes here a quaternion. Indeed, using Pauli matrices , set , , , then and . Moreover
This expression can obviously be identified with .
Our next comment concerns stability by renormalization. It has been shown  that some proposed constraints among masses of particles of the standard model may not be stable with respect to the renormalization group. This comment should be properly understood and maybe taken with a grain of salt. Indeed, the free parameters of the standard model are... free. Therefore one can renormalize them at will (at a given scale) and one can, in particular renormalize them in such a way that any relation between them is satisfied. Of course it is absolutely true that a renormalization prescription involves the choice of a scale (this ``substraction point'' is usually chosen equal to some value of where q is a four-momentum) and that a numerical relation between renormalized parameters may be deformed by a change of scale if the relation is not invariant under the renormalization group. But this fact does not mean that the relation itself is physically meaningless. For example, the relation which is valid in the on shell renormalization scheme of quantum electrodynamics can be imposed and is actually imposed (because it is experimenally true on shell). However this last relation is not invariant under the renormalization group equations of QED! The third and last comment to be made about these problems of relations between constants of the standard model was already made in the text (section 6) but we repeat it here. Descriptions of the Standard Model based on non commutative geometry supplemented by the choice of specific scalar products in the space of fields seem to lead, at the classical level, to relations between the - otherwise arbitrary - constants of the model. Gauge invariance alone allows for more freedom; in absence of a full description of (spontaneously broken) quantum gauge field theories in terms of non commutative geometry, such constraints should be considered, in our opinion, as educated guesses. The reader may refer to [18, 19] for a detailed analysis of these constraints in the Conne's framework. For us, the true power of the non commutative geometry desription of the standard model (and of quantum physics in general) is not tied up with the relevance of such constraints.
In its simplest ``version'', the standard model does not incorporate right neutrinos. From the point of view of non commutative geometry and if one restricts oneself to the leptonic sector (take for instance the example of one family), lacking right neutrinos is a little bit of a nuisance. Indeed, in such a case, and in the language of A.Connes , the bundle of leptonic species is non trivial and one needs to introduce a projector in the formalism, projector whose curvature itself enters the final expression. In the formalism explained in , the same phenomena appears because our approach (based on the tensorization of two by two complex matrices by arbitrary ones) leads to even dimensional square matrices. In order to accomodate an odd number of Weyl fermions (for instance and ) one has to embedd the odd dimensional matrix describing the connection into a even dimensional one by adding line an columns of zeros. But then action of the d operator creates non zero entries in such places. The curvature is then not equal to but to where p projects on the odd dimensional subspace spanned by the Weyl fermions. The result is, as it should, the usual standard model without right neutrinos. However, introducing right neutrinos in the game (like in  and like in section 5 of the present paper) simplifies considerably the formalism because one does not have to introduce such a projector. One cannot not say that ``Non commutative geometry predicts that the neutrino has a mass'' but it is clear that, from our perspective, the formalism is much simpler with a right neutrino than without. Let us remind the reader that such neutrinos are absolutely compatible with present experimental data since the that one introduces for each family is not coupled to the (transverse part of the) gauge fields. Its main interest is to give a mass to the corresponding particle (hence a Dirac spinor) and to introduce mixing between fermionic families via a fermionic analogue of the Kobayashi-Maskawa matrix. Introduction of right neutrinos in the Connes' formalism was recently investigated in .
We would like to thank our friends and colleagues at C.P.T. (in particular G. Esposito Farese) and at the University of Mainz (in particular R. Haussling and F. Scheck) for many discussions on those topics. One of us (R.C.) wants to thank the Erwin Schrodinger Institute, in Vienna, for its support and hospitality.