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Centre de Physique Théorique - CNRS - Luminy, Case 907 F-13288 Marseille Cedex 9 - France

Comments about Higgs fields, noncommutative geometry and the standard model


G. Cammarata
R. Coquereaux


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: Keywords: Higgs, standard model, electroweak interactions, non commutative geometry

tex2html_wrap_inline652 Work supported in part by a PROCOPE project between Mainz University and CPT Marseille-Luminy.

tex2html_wrap_inline654 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)

1. Introduction

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:

We also make a number of comments that may help the reader to see what is going on in this field.

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 [4] but can also be read independently; it should not be considered as an expository lecture on the approach initiated by [1].

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.

From an 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 calculus

A branch of non commutative differential geometry is non commutative differential calculus. The aim is to be able to consider objects like df or tex2html_wrap_inline658 , 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 tex2html_wrap_inline662 model studied in [4], 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 [1] 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 tex2html_wrap_inline664 with two elements that we call L and R. Call x the coordinate function tex2html_wrap_inline672 and y the coordinate function tex2html_wrap_inline676 . Notice that xy=yx=0, tex2html_wrap_inline680 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 tex2html_wrap_inline688 generated by x and y can be written tex2html_wrap_inline694 (where tex2html_wrap_inline696 and tex2html_wrap_inline698 are two complex numbers) and can be represented as a diagonal matrix tex2html_wrap_inline700 . One can write tex2html_wrap_inline702 and is isomorphic with tex2html_wrap_inline704 . We now introduce a differential tex2html_wrap_inline706 satisfying tex2html_wrap_inline708 , tex2html_wrap_inline710 and the usual Leibniz rule, along with formal symbols tex2html_wrap_inline712 and tex2html_wrap_inline714 . It is clear that tex2html_wrap_inline716 , the space of differentials of degree 1 is generated by the two independent quantities tex2html_wrap_inline720 and tex2html_wrap_inline722 . Indeed, the relation x+y=1 implies tex2html_wrap_inline726 , the relations tex2html_wrap_inline728 and tex2html_wrap_inline730 imply tex2html_wrap_inline732 , therefore tex2html_wrap_inline734 and tex2html_wrap_inline736 . This implies also, for example tex2html_wrap_inline738 , tex2html_wrap_inline740 , tex2html_wrap_inline742 , tex2html_wrap_inline744 etcMore generally, let us call tex2html_wrap_inline746 , the space of differentials of degree p; the above relations imply that a base of this vector space is given by tex2html_wrap_inline750 . Call tex2html_wrap_inline752 and tex2html_wrap_inline754 . This space tex2html_wrap_inline756 is an algebra: We can multiply forms freely but one of course has to take into account the Leibniz rule, for instance tex2html_wrap_inline758 . Since each tex2html_wrap_inline746 is two dimensional we can easily represent it in terms of matrices. More precisely, we can represent the element tex2html_wrap_inline762 as the diagonal matrix tex2html_wrap_inline700 and the the element tex2html_wrap_inline766 as the off diagonal matrix tex2html_wrap_inline768 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 tex2html_wrap_inline756 . Indeed, the relations


imply that the above representation using tex2html_wrap_inline774 matrices is indeed a homomorphism of tex2html_wrap_inline776 graded algebras from the algebra of universal forms tex2html_wrap_inline756 (graded by the parity of p) to the algebra of tex2html_wrap_inline774 complex matrices (with tex2html_wrap_inline776 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 tex2html_wrap_inline756 . Notice that the algebra tex2html_wrap_inline756 is infinite dimensional (since p ranges from 0 to infinity) but if we represent the whole of tex2html_wrap_inline756 in terms of tex2html_wrap_inline774 matrices acting on a fixed 2-dimensional vector space, the p grading is lost and only the tex2html_wrap_inline804 grading is left. The differential tex2html_wrap_inline706 obeys the usual Leibniz rule when it acts on elements of tex2html_wrap_inline688 but a graded Leibniz rule when it acts on elements of tex2html_wrap_inline756 , namely tex2html_wrap_inline812 where tex2html_wrap_inline814 denotes 0 or 1 depending if tex2html_wrap_inline820 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 tex2html_wrap_inline716 . Take tex2html_wrap_inline824 . The matrix representation of A reads therefore


The corresponding curvature is then tex2html_wrap_inline830 , but tex2html_wrap_inline832 and tex2html_wrap_inline834 , so that the curvature can also be written


We now chose a hermitian product on tex2html_wrap_inline756 by declaring the base tex2html_wrap_inline840 to be orthonormal. Then tex2html_wrap_inline842 . 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 [4] where it is written in the language of tex2html_wrap_inline774 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, tex2html_wrap_inline716 is of dimension 6 and tex2html_wrap_inline852 of dimension 12. If we take q points the dimension of tex2html_wrap_inline746 is tex2html_wrap_inline860 . 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 tex2html_wrap_inline716 can be defined as functions A(x,y) of two variables on X such that A(x,x)=0 and that elements of tex2html_wrap_inline852 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 tex2html_wrap_inline664 - that is our main example in the present paper - we recover the fact that an element A of tex2html_wrap_inline716 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 tex2html_wrap_inline774 matrices indexed by L and R. An element F of tex2html_wrap_inline852 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 tex2html_wrap_inline904 for all p explains that we can use a representation of fixed dimension (namely tex2html_wrap_inline774 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 section.

5. What are the Higgses ? The Yukawa interaction term

Higgs fields ( tex2html_wrap_inline924 ) allow left and right fermions ( tex2html_wrap_inline926 ) to communicate. In four dimensional Minkowski space, this is clear from the trilinear Yukawa couplings such as tex2html_wrap_inline928 that appear in the Lagrangian density of the Standard Model. This should be contrasted with terms like tex2html_wrap_inline930 or tex2html_wrap_inline932 where tex2html_wrap_inline934 or tex2html_wrap_inline936 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 tex2html_wrap_inline958 and similarly for quarks of charge -1/3 with tex2html_wrap_inline962 The tex2html_wrap_inline964 complex matrices tex2html_wrap_inline966 and tex2html_wrap_inline968 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 tex2html_wrap_inline980 and all right-handed quark fields into a single spinor tex2html_wrap_inline982 . Here tex2html_wrap_inline984 and tex2html_wrap_inline986 . The above Yukawa interaction term reads




The mass term is obtained by shifting tex2html_wrap_inline992 and tex2html_wrap_inline994 by a real constant with dimension of a mass that we call tex2html_wrap_inline996 . The mass term itself is therefore described by the mass matrix


Writing tex2html_wrap_inline1000 , the whole fermionic lagrangian, for quarks, reads tex2html_wrap_inline1002 with


Where tex2html_wrap_inline1006 and tex2html_wrap_inline1008 collectively refer to those components of the gauge fields coupled to the left and right handed sectors and tex2html_wrap_inline1010 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 tex2html_wrap_inline966 and tex2html_wrap_inline968 by tex2html_wrap_inline1016 and tex2html_wrap_inline1018 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 tex2html_wrap_inline1024 and similarly for the neutrinos tex2html_wrap_inline1026 The tex2html_wrap_inline964 complex matrices tex2html_wrap_inline1016 and tex2html_wrap_inline1018 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 tex2html_wrap_inline1036 , with tex2html_wrap_inline1038 and where both tex2html_wrap_inline1040 and tex2html_wrap_inline1042 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 tex2html_wrap_inline1048 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 tex2html_wrap_inline1050 appears as the inverse of a quantity encoding a discrete set of fundamental lengths. The second piece tex2html_wrap_inline688 is a generalized connection: it incorporates both Yang-Mills and Higgs fields.

6. The bosonic lagrangian

The theory of - usual - connections explains why tex2html_wrap_inline1054 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 tex2html_wrap_inline1056 is the natural object (curvature) associated with the Higgs field tex2html_wrap_inline924 introduced in the section 4 and defined as a non local connection on the discrete set tex2html_wrap_inline664 .

Now, we do not have a discrete set but a space tex2html_wrap_inline1062 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 tex2html_wrap_inline1068 associated with the generalized connection tex2html_wrap_inline688 introduced in the previous paragraph is






The symbols tex2html_wrap_inline1078 and tex2html_wrap_inline1080 denote the usual curvatures of Yang-Mills fields associated with hermitian fields L and R. The expression of matrix elements of tex2html_wrap_inline1068 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 [1]. 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 [2] (replacement of the so-called algebra tex2html_wrap_inline756 by tex2html_wrap_inline1090 ). A detailed exposition of the formalism of [2] using K-cycles and Dixmier trace can now be found in several places [3, 17, 18]. The matrix elements of tex2html_wrap_inline1068 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 tex2html_wrap_inline1068 have a dimension of a mass squared and that, as a consequence, an arbitrary mass scale tex2html_wrap_inline996 appears in the formula. Explicitely, the term tex2html_wrap_inline1098 and its adjoint can be computed from the expressions of tex2html_wrap_inline1010 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 tex2html_wrap_inline1102 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 tex2html_wrap_inline996 in the Higgs fields tex2html_wrap_inline992 and tex2html_wrap_inline994 (see [4] 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 tex2html_wrap_inline1110 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 tex2html_wrap_inline1112 , tex2html_wrap_inline1114 and tex2html_wrap_inline1116 where g, tex2html_wrap_inline1120 , tex2html_wrap_inline1122 and tex2html_wrap_inline696 are undetermined. If, on the other hand, we decide to introduce a tex2html_wrap_inline1126 constant tex2html_wrap_inline1128 in front of tex2html_wrap_inline1110 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 tex2html_wrap_inline1128 , we can rescale gauge fields as usual by tex2html_wrap_inline1134 and also the Higgs fields by tex2html_wrap_inline1136 . Under identification with the usual lagrangian one obtains immediately tex2html_wrap_inline1138 ; 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 tex2html_wrap_inline1140 is not modified but the second relation becomes tex2html_wrap_inline1142 . Moreover, as we shall see below, the value of tex2html_wrap_inline1120 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 tex2html_wrap_inline1148 . The mass value for the Higgs particle coming from this usual expression is tex2html_wrap_inline1112 . Notice also that the left hand side contains no additive constant (absence of cosmological term).

In our case, the Higgs potential itself coming from tex2html_wrap_inline1152 reads,


If we now express tex2html_wrap_inline1010 in terms of the component Higgs fields and in terms of the matrices of Yukawa coupings then remove the factor tex2html_wrap_inline1128 , in front, by rescaling the fields, we see that tex2html_wrap_inline1160 contains a term equal to tex2html_wrap_inline1162 but the term tex2html_wrap_inline1164 leads to a kinetic term for tex2html_wrap_inline1166 equal to tex2html_wrap_inline1168 so that the mass of the Higgs field does not depend on the mass of fermions and stays undetermined (remember that tex2html_wrap_inline1122 is a free parameter). Other authors [18], using a different formalism find quite stringent constraints relating tex2html_wrap_inline1140 to the fermionic masses.

The full bosonic interaction contains also a term tex2html_wrap_inline1174 ; using the previous expression for tex2html_wrap_inline1176 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 tex2html_wrap_inline1192 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 tex2html_wrap_inline1200

would lead to x = 22 / 9 and tex2html_wrap_inline1204

For leptons alone, the normalization tex2html_wrap_inline1206

would lead to x = 6 and tex2html_wrap_inline1210

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 tex2html_wrap_inline1220 (or tex2html_wrap_inline1222 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 tex2html_wrap_inline1228 but a group metrically isomorphic with the tex2html_wrap_inline1230 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 tex2html_wrap_inline1220 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 tex2html_wrap_inline1068 we obtain a term tex2html_wrap_inline1242 that gives a mass to the W and the Z. The trace itself reads




This gives the relation tex2html_wrap_inline1252 which is well known in the standard model. In general, we have tex2html_wrap_inline1254 and this becomes only a constraint (namely tex2html_wrap_inline1256 ) if we set tex2html_wrap_inline696 to the ``natural'' value tex2html_wrap_inline1260 as discussed before. One could hope that such relations could hold at a scale where the previous value for tex2html_wrap_inline1120 is experimentally satisfied (maybe at some grand unification scale). Notice that other authors [18], 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 [6]) 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 tex2html_wrap_inline1264 lives is naturally tex2html_wrap_inline804 graded by L and R, i.e. tex2html_wrap_inline1264 can be decomposed into a left and a right part. Therefore transformations that map tex2html_wrap_inline1264 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 tex2html_wrap_inline804 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 tex2html_wrap_inline1304 . 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 tex2html_wrap_inline1316 . 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 tex2html_wrap_inline1326 -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 tex2html_wrap_inline1326 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 tex2html_wrap_inline1326 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 tex2html_wrap_inline1326 '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 tex2html_wrap_inline1038 . 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 tex2html_wrap_inline1338 and B; they are denoted, as usual, by tex2html_wrap_inline1342 and Y. The last four are matrices that appear as coefficients of the Higgs fields tex2html_wrap_inline1346 and tex2html_wrap_inline994 ; they give rise (after having added the hermitian conjugate) to the Yukawa and mass interaction term. We call them tex2html_wrap_inline1350 , tex2html_wrap_inline1352 , tex2html_wrap_inline1354 and tex2html_wrap_inline1356 . More precisely, consider the following (block) matrices:


where a , b, g, e are themselves square matrices, for example of size tex2html_wrap_inline964 if we consider only quarks coming in 3 families. In this case, we decide to label the basis as follows: tex2html_wrap_inline1366 . Let us define tex2html_wrap_inline1368 , tex2html_wrap_inline1370 and tex2html_wrap_inline1372 . The electric charge is tex2html_wrap_inline1374 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 tex2html_wrap_inline756 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: tex2html_wrap_inline1386 with tex2html_wrap_inline1388 , tex2html_wrap_inline1390 and tex2html_wrap_inline1392 These constraints are satisfied if and only if, on top of the relation tex2html_wrap_inline1394 , the matrices e,b,g,a satisfy also the relations tex2html_wrap_inline1398 , tex2html_wrap_inline1400 and tex2html_wrap_inline1402 Indeed, one finds tex2html_wrap_inline1404 , tex2html_wrap_inline1406 and tex2html_wrap_inline1408 . This imply in particular g a = a g and e b = b e. One can then check that matrices tex2html_wrap_inline1414 and tex2html_wrap_inline1416 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 tex2html_wrap_inline1418 ).

Define now tex2html_wrap_inline1420 and write tex2html_wrap_inline1422 This expression can not be real, indeed tex2html_wrap_inline1424 would imply tex2html_wrap_inline1426 , tex2html_wrap_inline1428 , but the other constraints would lead to a contradiction ( tex2html_wrap_inline1430 ). To obtain a real expression, one has to add tex2html_wrap_inline1432 and tex2html_wrap_inline1434 . Writing tex2html_wrap_inline1436 gives


and we recognize the expression of tex2html_wrap_inline1440 given in section 5, with the identification tex2html_wrap_inline1442 and tex2html_wrap_inline1444 . Warning: The matrix tex2html_wrap_inline1446 defined previously in terms of the tex2html_wrap_inline756 matrices is not equal to the matrix tex2html_wrap_inline1010 defined in section 5; in order to compare the two expressions, one has to first add the conjugated expressions tex2html_wrap_inline1432 and tex2html_wrap_inline1434 . Taking into account the constraints on blocks a,g,e and b, one obtains the relations: tex2html_wrap_inline1460 and tex2html_wrap_inline1462 These relations do not imply any ``new'' constraints on mass matrices tex2html_wrap_inline968 and tex2html_wrap_inline966 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 [6] 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 [14] for the case of three families.

Remark: The quantity tex2html_wrap_inline1432 may be thought as the contribution to the lagrangian of a particular representation of Sl(2|1). One can think of tex2html_wrap_inline1434 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, tex2html_wrap_inline1432 is not hermitian but tex2html_wrap_inline1434 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 tex2html_wrap_inline1488 and tex2html_wrap_inline1490 as before, one can define the following ``hatted'' tex2html_wrap_inline756 matrices: tex2html_wrap_inline1494 and tex2html_wrap_inline1496 It is then straightforward to check that these hatted tex2html_wrap_inline756 matrices generate (thanks to the same commutation relations) matrices tex2html_wrap_inline1500 , tex2html_wrap_inline1502 and tex2html_wrap_inline1504 , with, for example tex2html_wrap_inline1506 We obtain in this way a new representation (the relation tex2html_wrap_inline1508 being automatically satisfied since tex2html_wrap_inline1510 ). With tex2html_wrap_inline1432 as before, we can rewrite tex2html_wrap_inline1434 as tex2html_wrap_inline1516 and tex2html_wrap_inline1518 as


so that tex2html_wrap_inline1518 itself appears as the contribution associated with the ``hatted'' representation. If one wishes to use tex2html_wrap_inline1434 in terms of a contribution of antiparticles, for instance tex2html_wrap_inline1526 as tex2html_wrap_inline1528 , 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: tex2html_wrap_inline1038 with tex2html_wrap_inline1532 , tex2html_wrap_inline1534 and define matrices Omega as previously, in terms of new tex2html_wrap_inline964 block matrices e,b,g,a. However, in the case of leptons, the constraints for the hypercharge generator are different. Indeed, tex2html_wrap_inline1540 with tex2html_wrap_inline1542 , tex2html_wrap_inline1544 and tex2html_wrap_inline1546 These constraints are satisfied if and only if, on top of the relation tex2html_wrap_inline1394 (which ensures that commutation relations for SL(2|1) hold), the matrices e,b,g,a satisfy also the relations tex2html_wrap_inline1554 and ag = 0. With these constraints, one can then check that matrices tex2html_wrap_inline1414 , tex2html_wrap_inline1416 and Y defined as before in terms of the matrices tex2html_wrap_inline756 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 tex2html_wrap_inline1418 ).

Again, we define tex2html_wrap_inline1420 and write tex2html_wrap_inline1422 This expression can not be real, and, in order to obtain a real expression, one has to add, as before, tex2html_wrap_inline1432 and tex2html_wrap_inline1434 . Writing tex2html_wrap_inline1436 gives


We recognize the expression of tex2html_wrap_inline1440 given in section 5, with the identification tex2html_wrap_inline1582 and tex2html_wrap_inline1584 but the matrices a,g,e and b are not totally arbitrary since they should here satisfy the constraints tex2html_wrap_inline1554 and a g = 0. These relations do not imply any constraints on mass matrices tex2html_wrap_inline1018 and tex2html_wrap_inline1016 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 ( tex2html_wrap_inline964 ) matrices tex2html_wrap_inline1608 and choose a,g,e and b in such a way that tex2html_wrap_inline1614 , tex2html_wrap_inline1616 , tex2html_wrap_inline1618 and tex2html_wrap_inline1620 , then, all commutation relations for tex2html_wrap_inline756 matrices are still satisfied. The generators tex2html_wrap_inline1624 and tex2html_wrap_inline1626 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 tex2html_wrap_inline1630 . 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 [15] (take tex2html_wrap_inline1608 nilpotent matrices). Relation between family mixing and existence of such representations was suggested in [6] 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 tex2html_wrap_inline1446 and tex2html_wrap_inline1432 as before and added the (usual) complex conjugate, one obtain a real expression and one can choose to diagonalize simultaneously tex2html_wrap_inline1416 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 tex2html_wrap_inline1608 . 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 tex2html_wrap_inline804 -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 tex2html_wrap_inline1038 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.


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.

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Next: References

Robert Coquereaux
Thu Oct 24 15:18:22 MET DST 1996