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Lectures on Unification

In these lectures we review the motivation, principles of and (circumstantial) evidence for the program of unification of the fundamental forces. In an appendix, we review the group theory pertinent to the program.


Introduction
The standard model of the electromagnetic, weak and strong interactions is the corner stone of elementary particle physics [1,2]. It is a lagrangian field theory of quark, lepton and gauge bosons degrees of freedom with the spontaneous breakdown of electroweak symmetry achieved by an elementary higgs scalar potential. While the standard model is enormously successful at present day energies, it is likely to be the "low"-energy limit of a more complete and perhaps simpler description of these interactions -a description which derives from the experience of the standard model, in the sense of being a lagrangian field theory, being a gauge theory and which uses the key concepts of spontaneous symmetry breaking. Grand unification [3,4,5,6], where in the standard model gauge symmetry is embedded in larger symmetries is such a program and is a subject of the present set of lectures. Another key unification is that of particle types, viz., particles of bosonic and fermionic types. Supersymmetry [7,8,9] is the symmetry that treats these degrees of freedom on the same footing and may be combined with lagrangian field theory.
In particular, modern approaches to unification simultaneously require grand unification as well as unification of bosonic and fermionic statistics and is called supersymmetric unification and is the framework within which the present discussion will take place. These symmetries, however, must be broken or hidden since there is no (direct) evidence for such unification.
In these lectures we will review the motivation, principles and circumstantial evidence for the program of unification of the fundamental forces, with the exception of the gravitational forces.
The aim of the lectures at this school is to bring the participants up to date with the current status of research in the areas covered at the school assuming as little as possible. We will mention virtually all the central notions that enter the construction of the unification program, in italics. We note, however that many of the preliminaries are already presented in standard textbooks [10,11] and we will frequently refer the reader to them for tracing the primary sources.
The relevant group theory is presented in an appendix and is a summary of results discussed elsewhere [12,13].

The Standard Model
At all length scales probed thus far at high energy accelerators, there has been no evidence to suggest that the fundamental constitutents of matter, namely the quarks and the leptons are Finally the strong interactions mediated by the massless gluons, g rendered short ranged by a yet to be discovered mechanism for the confinement of colour quantum number that is carried by the gluon (indeed, as it is by the quarks). The gluons are the gauge bosons of the underlying SU (3) colour gauge group and eight fields have to be introduced corresponding to the number of infinitesimal generators. The quarks come in three colors and transform as triplets under the color gauge group, whereas the leptons are singlets under this gauge group and do not participate in the strong interactions at the tree-level. [In the following we will be suppress the color indices assuming that they are correctly summed over.] The manner in which particles are coupled to the gauge fields is dictated by which representation of the relevant gauge groups they lie in. The principle of gauge invariance also dictates the manner in which particles interact between themselves. Only those couplings between the particles are allowed which are left invariant by the action of a gauge transformation. This picture, thus, requires us to specify the transformation properties of the matter fields under the gauge group SU (2) × U (1). In particular, the left-handed projections of the u and d quarks, T transforms as a doublet under SU (2) and carries the hypercharge 1/3, whereas the the right handed projections u R and d R transform as singlets under SU (2) and carry hypercharges 4/3 and −2/3 respectively. Mathematically this would correspond to a term in the Lagrangian density that would look like: Analogously we have the lepton doublets [ν L e L ] T which tranform as a doublet and with hypercharge −1 whereas the right-handed projections ν R and e R transform as singlets and carry hypercharge of 0 and −2 respectively. We note here that the right handed neutrino is completely inert with respect to the standard model gauge group and may even be left out of the spectrum.
A consistent picture arises when the electromagnetic and weak interactions are considered simultaneously in an electroweak framework based on a group SU (2) × U (1) [where SU (2) (or more generally SU (N ) is the group of 2 × 2 (or more generally N × N ) unitary matrices] which is then broken spontaneously by the Higgs mechanism when a standard model Higgs doublet of scalar fields φ T = [φ + φ 0 ] T is introduced to produce U (1) of electromagnetism, and in the process turns three of the gauge bosons, now named W ± and Z 0 , massive. The higgs mechanism occurs when a quartic potential is introduced for the doublet and when the classical potential turns into one by the arrangement of specific relations between the mass parameter and the quartic coupling wherein the ground state is the asymmetric minimum. More precisely the higgs potential is written down as: These transformation properties then specify the nature of the kinetic energy terms of the standard model particles, as we saw for the quarks, leptons and the higgs fields. Finally the kinetic energy terms of the gauge bosons themselves: For the non-abelian gauge fields the kinetic energy involves self-couplings of the gauge fields, a feature not present in electrodynamics. This has a crucial implication for the strong interactions -aymptotic freedom is a property arising from this feature. We must also note that the parameters in the Lagrangian above themselves are "running" coupling constants whose evolution is governed by the renormalization group equations. In particular, for the gauge couplings, in the standard model, we have the one-loop evolution equations for the couplings: where Q is the momentum and F is the number of families. We note here that the the quadratic Electroweak symmetry is broken when µ 2 is chosen negative with λ > 0. In particular, it is possible to arrange the parameters to yield the vacuum expectation value to the electrically neutral component: It is then possible to work through the details of the higgs mechanism to produce expressions for the masses of the W ± and Z bosons: One electrically neutral scalar higgs boson is left behind after spontaneous symmetry breaking.
Furthermore, from experiment, we have the relation for v in terms of the Fermi constant G f = The transformation properties also constrain the interactions between the particles themselves; the term that we add to the Lagrangian must be invariant under the combined gauge group. Such terms imply Yukawa couplings between the higgs doublet and left-and right-handed matter fields. Gauge invariance allows terms of the type: When spontaneous symmetry breaking occurs then the vacuum expectation value of the Higgs field multiplied by the Yukawa coupling gives rise to an effective mass term for the standard model fermions. Note that the absence of the right handed neutrino implies that the neutrino is massless in the standard model. Furthermore, at the perturbative level, the absence of lepton (and baryon) number violating couplings rules out the possibility of Majorana type masses which can be added to the lagrangian to provide a mass to the left-handed neutrino. We also note that we would finally have to sum over all the families. Since all the quarks have non-zero masses, once the electroweak symmetry is broken, the quarks may mix amongst themselves, viz., that the "flavor" eigenbasis would now not correspond to the "mass" eigenbasis. This would then be accounted for in the standard model by the Cabibbo-Kobayashi-Maskawa mechanism which is also rich enough to contain a single CP violating phase. We do not discuss this any further except to note that the standard model falls into the class of "milliweak" CP violating models which is yet to be confirmed experimentally.

SU(5)
Earliest examples of grand unification were provided by those based on the groups SU (4) × respectively. These may easily seen to be precisely the quantum numbers of one standard model family. In particular, they corresond to the left handed lepton doublet, right handed down-type quark singlet (conjugate), the right handed electron (conjugate), the right handed up-type quark singlet (conjugate) and the left handed quark doublet, respectively. Among other things, this would imply that transitions are possible between quark and lepton states [proton decay problem] and mass relations between various fermions, now unified into irreducible representations of the groups.
It may also be seen that a 5-dimensional scalar multiplet can accomodate the electroweak doublet but the electroweak singlet, colored triplet must be very massive in order to prevent rapid proton decay [14]. The 24 dimensional representation may also be considered, with the The theory is specified by writing down the terms in the Lagrangian that couple these fields.
In particular, we see that Yukawa couplings may be written down for the fermions in the 5 and 10 and the 5 dimensional scalar. Indeed, one may then compute the tensor products of these irreducible representations and find in the sum of irreducible representations a piece that is invariant (i.e., a singlet) under SU (5).

Charge Quantization
The fact that standard model fermions of differing hypercharges are accomodated into irreducible representations of SU (5) implies there is a basis for relating the hypercharge assignments of those fermions that are in the same multiplet. For instance, when we consider the electroweak doublet and the down-type anti-quark that lie in the same 5 it implies that the action of the same diagonal hypercharge generator produces eigenvalues of their respective hypercharges.
This in turn implies that charge is now quantized. Furthermore, we have the result that the normalization of the hypercharge generator is now related to the normalization of the diagonal generators of SU (2) and SU (3).
The seemingly arbitary choice of gauge couplings in the standard model would also have to be replaced by a unique gauge coupling in the event of unification into SU (5). However, we must first fix the normalization of the hypercharge generator of the standard model, vis a vis the generator that is embedded in SU (5). We recall the relations: In the standard model, we have the Gell-Mann-Nishijima type relation: However, in SU (5), if we consider the SU (3) c subgroup to lie in the upper 3 × 3 diagonal sub-group and SU (2) (weak-isospin) to lie in the lower 2 × 2 diagonal sub-group, then T 3 = diagonal(0 0 0 1 − 1)/2 and the hypercharge would be proportional to ). If we have to correctly produce the electric charge assignments to the 5, then we would have to define Q = T 3 + 5/3Y ′ . This then gives us the required boundary condition that g ′ = 3/5g at the unification scale.

Coupling Constant Unification
A unification scale M G ∼ 10 16 GeV is suggested by gauge coupling unification, above which physics would be described by a grand unified theory [3] based on a gauge group G. Indeed, the arrival at the structure of fundamental interactions from renormalization group flow has a predecessor in the example of asymptotic freedom in deep inelastic scattering experiments and thus gauge coupling unification is an extremely encouraging sign that grand unified theories are the right step for a theory of fundamental interactions. The evolution equations we consider are precisely those we encountered earlier eq.(4). These equations provide the following system of relations between the two inputs at low-energies α and α s and the unification scale, the value of the unified coupling constant α G and the value of sin 2 θ w at low-energies.

Complexity of Representations
In shown that the only groups that admit complex representations are SU (n), n ≥ 3, SO(4n + 2) and E 6 .

SO(10) and Anomaly Cancellation
The seemingly arbitary assignments of a standard model fermion to representations of SU (5) finds a natural resolution when we consider an even larger gauge symmetry, viz., SO (10). It may be easily seen from the Dynkin diagram structure of the algebra of SO(10) that SU (5) is a subalgebra, with SU (5) × U (1) being a maximal subalgebra. We may either choose the SU (5) as it stands as the Georgi-Glashow SU (5)

Neutrino Masses
Note that whereas in the standard model, the field content forbids a Dirac mass for the neutrinos since the right handed neutrino is absent and Majorana mass is forbidden by the conservation of lepton number. In grand unified models, neither of these principles is respected and a wide variety of possibilities exists for the generation of neutrino masses. However, far from being arbitrary, it should be possible to uncover information regarding the structure of unified theories from accurate determination of small and eventually large neutrino masses and mixing angles, viz., neutrino masses may be viewed as bearing an imprint on the structure of grand unification and the nature of the breakdown of unification [15].
One pedagogical example we consider is one wherein the right-handed neutrino receives a

Hierarchy Problem
The presence of disparate scales in the theory, M G and the weak scale M W ∼ 174 GeV, expected to be separated by more than ten orders of magnitude, would render the mass of the Higgs scalar of the electro-weak model ∼ M W , unnatural-natural. Should the Higgs scalar be elementary, then one manner in which it would remain naturally at the weak scale is due to cancellation of divergences as in supersymmetric unified models [7,8]. This is further discussed in the next section.

Supersymmetric unification
This section is extracted from a recent review article [16] and is sufficiently detailed to serve as a self-contained discussion of the subject. In what follows we recall some of the essential successes of the recent investigations [17] in the theory of supersymmetric unification. This was spurred by the confrontation of the ideas of unification by the precision measurements of the gauge couplings of the standard model at the LEP [18]. A highly simplified understanding of this feature may be obtained from a glance at the one-loop evolution equation for the standard model gauge couplings, more correctly the gauge couplings of the minimal supersymmetric standard model assuming that the effective supersymmetry scale is that of the weak scale, with t = ln µ: where we have assumed three generations. One may then integrate these equations to obtain: in the experimental range [22].
Besides the vindication of top-quark discovery predicted by susy guts, another strong test takes shape in the form of its prediction for the scalar spectrum. In the MSSM the mass of the lightest scalar is bounded at tree level by M Z since all quartic couplings arise from the D-term in the scalar potential. The presence of the heavy top-quark enhances the tree-level mass, but the upper bound in these models is no larger than 140 GeV.
Other predictions for softly-broken susy models arise when a detailed analysis of the evolution equations of all the parameters of the model are performed and the ground state carefully analyzed. In the predictive scheme with SO(10) unification, the model is further specified by M 1/2 , m 0 and A, the common gaugino, scalar and tri-linear soft parameters [8]. It turns out that in this scheme M 1/2 is required to come out to be fairly large, at least ∼ 500 GeV implying a lower bound on the gluino mass of a little more than a TeV and providing a natural explanation for the continuing absence of observation of susy particles from scenarios based on radiative electro-weak symmetry breaking [23]. [An extensive study of the NMSSM with SO(10) conditions has also been performed [24].] Considerably greater freedom exists when the SO(10) boundary condition is relaxed [25]. In summary many predictions and consistency of the MSSM and its embedding in a unified framework have been vindicated; however it is important to continue theoretical investigations and checks to the consistency of these approaches and extensions to include the lighter generations [26].

Gravitation
The final frontier that still remains to be explored is a framework within which a consistent incorporation of the gravitational interactions is successful. Whereas it has not been possible to replace the Einstein theory by a quantum version due to bad ultra-violet behaviour, supergravity possesses improved ultra-violet properties [7]. String theories [27] often contain supergravity in their low energy spectrum and as a result supersymmetric unification is a favored candidate for these reasons as well.
Note that the group of 2 × 2 unitary matrices is obtained by Furthermore, the existence of continuous symmetries in field theory implies the existences of conserved currents.
It is customary to define the combinations T ± = T 1 ± T 2 , which are the familiar raising and lowering operators. It is possible to diagonalize only one of the operators T i and customarily it is chosen to be T 3 . In terms of these redefined operators, the commutation relations now read: Furthermore, one defines the quadratic operator T 2 = (T + · T − + T − · T + )/2 + T 2 3 . It is also well know that one may define a basis for higher angular momentum states as an eigenstate of T 2 and T 3 in terms of the quantum numbers (j, m) with −j ≤ m ≤ j and the state is then (2j + 1)-degenerate. This is an example of a higher dimensional representation of the angular momentum algebra. One may string out the entire (2j + 1)-dimensional basis vectors on a line and the lowering and raising operators cause transitions between these states.
SU (3) is the smallest algebra which shows a structure rich enough to be extended to the remainder of the semi-simple algebras namely the classical series and the exceptional series.
This algebra may be defined in terms of 8 linearly independent operators. Conventionally these 8 may be represented by the Gell-Mann matrices. Of these, two may be simultaneously diagonalized. These simultaneously diagonalizable which are called T z and Y operators span the Cartan sub-algebra of the original algebra. These two are proportional to the two diagonal Gell-Mann matrices λ 3 and λ 8 respectively. The remaining 6 operators are named T ± , V ± and U ± and are equal to (λ 1 ± iλ 2 )/2, (λ 4 ± iλ 5 )/2 and (λ 6 ± iλ 7 )/2 respectively. We stick to these choices of linearly independent operators since they naturally generalize the raising and lowering operators of the SU (2) algebra.
One may then work out the commutation relations between the 8 linearly independent operators knowing their representations in terms of the Gell-Mann matrices. For instance, we may list the commutation relations enjoyed by T + with system of operators we have chosen: Generalization to other algebras may be performed by considering the operators that lie in the Cartan-subalgebra and the remainder broken up into (dim G -dim H) root vectors.
It is now possible to associate to the adjoint representation the Killing form: It turns out that for our choice of linearly independent vectors, the only non-zero answers occur In short, we have the root vectors α i (k), i = 1, 2, 3 of the algebra with roots ±a, ±(−a/2 + b), ±(a/2+b), respectively when we chose the vector in the Cartan subalgebra k = at z +bY. [Note that α 3 = α 1 + α 2 .] Corresponding to these three roots are the vectors in the Cartan subalgebra h α i , i = 1, 2, 3, h α 1 = t z /3, h α 2 = −t z /6 + y/4, h α 3 = t z /6 + y/4 such that α i (k) = (h α i , k). Now, we may define the scalar product on the space of roots with the definition: In particular, for the system of roots α i , we have < α i , α 1 >= 1/3, < α 1 , α 2 >= −1/6, < α 1 , α 3 >= 1/6, < α 2 , α 3 >= 1/6. This can be expressed geometrically as vectors of equal length 1/ √ 3, with α 1 and α 2 at an angle of 120 0 and α 1 and α 3 at an angle of 60 0 . These may be represented as non-orthogonal vectors in a two-dimensional plane. Generalization to higher algebras would entail the representation of roots in a space whose dimension is equal to dim H.
What we can observe from the eq.(17) and the definition of the Killing form and the structure of the roots and the associated root vectors, is that the commutation relation between root-vector e α of the root α and e −α of the root −α yields the element of the Cartan algebra h α multiplied by the Killing form (e α , e −α ), the commutation relation between a root-vector and an element of the Cartan algebra produces an eigenvalue equation for the same root-vector, where the eigenvalue is the root in question and finally, a commutation relation between two root-vectors yields an expression that is non-zero only if the sum of the two roots associated with the root-vectors is itself a root: [e α , e β ] = N αβ e α+β , if α + β is itself a root, or zero otherwise.
These properties may be simply generalized for a larger and more abstract (semi-simple) Lie algebra. However the generalization itself imposes severe restrictions on the nature of the rootspace. In order to discuss the generalization, we will first of all discuss the higher dimensional representations of the SU (3) algebra, having encountered thus far, the fundamental representation and the adjoint representation. A representation is obtained when we have for each element of the algebra a linear transformation (i.e., a matrix) on a vector space (i.e., column vectors) that preserves the commutation relations. Note that for members of the Cartan subalgebra we can simultaneously diagonalize the associated matrices and the column vectors φ a can be so chosen such that In the case of the fundamental representation of SU (3) Just as we have reflection symmetry about the orgin in SU (2) algebra, for larger algebras we have a richer symmetry structure which is known as the W eyl Group.
One may then use the defining properties of Lie algebras to deduce many of the properties of weight vectors in general. The multiplicity of states with a fixed weight may in principle exceed one. However for the adjoint representation it is unity for each root with the exception of the Cartan sub-algebra. Furthermore it turns out that for the root-system, the following identity has to be respected: < α, β > 2 < α, α >< β, β > = mn 4 (23) where m and n are integers [this follows from an important and interesting property of the roots that if α is a root, 2α cannot be a root]. However, the left hand side may be seen to be nothing but cos 2 θ where θ is the angle between the root-vectors α and β. This then implies that cos 2 θ can be 0, 1/4, 1/2 and 3/4. Now we describe further characteristics of the SU (3) root system which by now may have already become evident: while there are 6 roots, 3 or them are negatives of the other 3. Of these only two are linearly independent. One then considers a certain ordering of these roots to define the notion of a positive root. In the present case, these may seen to be α 1 , −alpha 2 and α 3 [simply put, these roots are the ones where the coefficient of a is positive when we consider the commutation relations of the root vectors with aT 3 + bY ]. Out of these, α 1 can be written as α 3 + (−α 2 ). Then we are led to the definition of a simple root as one that cannot be written as a sum of two positive roots. In the present example, −α 2 and α 3 are the simple roots.
All the information regarding the algebra can then be expressed in terms of the Cartan matrix. The Cartan matrix is defined as the matrix whose elements are given by In the case of SU Corresponding to each representation, one may define the Dynkin label of the representation Λ: Multiplication of the vector Λ i by the inverse of the Cartan matrix, which is known as the metric tensor on the root space, express the element of the representation as a linear combination of the simple roots.
Given the Dynkin label of the highest weight, one may then evaluate the dimensionality of the representation to which it belongs by use of the Weyl dimension formula which reads: where δ = ( α>0 α)/2.
The tensor product of irreducible representations breaks up into a sum of irreducible representations. In particular, for the SU (n) algebras, the Young's tableaux method allows one to compute the sum in a straightforward manner. For other algebras, there are methods to perform the computations and in particular, the Dynkin labels also allow one to figure out the product representations.
A final application of the Dynkin labels allow us to study the branching rules of a representation under its subalgebras. The branching rules for many interesting groups are catalogued in the primary sources.