Since mixing can be controlled by group theoretical arguments and we are finally interested in a lattice evaluation of the relevant matrix elements, we focus on the space-time symmetry of the hypercubic lattice in this section.
The following arguments are not affected by the positions of the covariant derivatives within the three-quark operators. For ease of notation we will therefore assume that all derivatives act on the last quark field. Moreover we want to construct linear combinations of the elementary three-quark operators in eq. (4.1) that have special symmetry properties. To this end we
4.1. THE SYMMETRY OF THE HYPERCUBIC LATTICE 53 Table 4.1: Symmetries of the hypercubic lattice.
e1 −e1 e2 −e2 e3 −e3 e4 −e4
introduce tensorsT(i)that represent the coefficients of these linear combinations. Suppressing the color indices and omitting the common space-time coordinate x, a local three-quark operator then generically reads:
O(i)=Tαβγµ(i)
1...µnfαgβDµ1. . . Dµnhγ. (4.3) In this way 43+n independent three-quark operators can be generated, which makes the com-plexity of the mixing problem increase with the number of derivatives. By appropriately choosing the coefficients T(i) an operator basis can be constructed, in which the general renormalization matrixZij takes a block diagonal form. Since then only operators belonging to the same block do mix, this procedure reduces mixing to a minimum.
The practical construction of this basis is based on the transformation properties of the three-quark operators under rotations and reflections in space-time. Three-three-quark operators that do not have equivalent transformation properties do not mix. This condition is fulfilled by any two operatorsO(i) andO(j) that belong to inequivalent irreducible representations of the associated symmetry group H(4). Therefore the related renormalization matrix elementsZij
and Zji vanish. It is obvious that grouping the space of three-quark operators into mixing subsets gives rise to the described decomposition of Z into a block diagonal matrix, where each block belongs to a set of identically transforming operators.
So the mixing problem is inherently linked to the knowledge of all identically transforming operators, which can in turn be read off from the H(4) irreducibly transforming multiplets of three-quark operators. We will therefore study the irreducible representations of the (spino-rial) hypercubic group in the following subsections.
The Hypercubic Group
Let us for the time being ignore the spinorial nature of the quark fields and introduce the symmetry group of the hypercubic lattice (see, e.g., [80]). This so-called hypercubic group H(4) defines, how objects with integer spin behave under transformations of the discretized space-time.
In terms of group theory, the hypercubic lattice can be thought of as a set of symmetry transformations of the lattice axes. Letej, j = 1, . . . ,4, denote unit vectors pointing in the direction of the four canonical axes and let us arrange them as shown in table 4.1. Then the symmetry group of a lattice consists of all transformations that leave the symmetry itself untouched, i.e., the lattice looks the same before and after the transformation. It turns out that two classes of operations fulfill this request. The first one is the interchange of two axes:
(ei,−ei)↔(ej,−ej). (4.4)
This corresponds to exchanging two rows in the table. As there is a total of four rows, it is readily seen that these operations represent the permutation group of degree four, S4. Inverting an axis is the other symmetry operation one can think of:
(ei,−ei)7→(−ei, ei). (4.5) In the diagram this is equivalent to flipping the two entries within one row, and the cor-responding symmetry is Z2. Taking into account all four rows one arrives at Z24. The full structure of the lattice symmetry group is found when working out the commutation relations between the two operations of exchanging and reflecting the axes. This way an isomorphism between the symmetry group of the hypercubic lattice and the semidirect product Z24oS4 (wreath product of Z2 and S4) is found. It is straight forward to read off the group order, i.e., the number of elements that constitute this finite group: 4!·24 = 384.
After this rather qualitative description of the symmetry structure we now turn to a more abstract approach. According to [81] the hypercubic group can be defined by the six generators t,γ,I1,..., I4 and a set of generating relations,
Ii2= 1, IiIj =IjIi, tI1=I1t, tI2=I4t, tI3=I2t, tI4=I3t, γI1=I3, γI2=I2γ, γI4=I4γ,
γ2= 1, t3= 1, (tγ)4= 1. (4.6)
Expressed in terms of the above introduced symmetry operations the generators Ij can be interpreted as inversions whereas t and γ represent a combined inversion and interchange of the axes. By definition each of the 384 elements G∈H(4) can be expressed as a product of these six generators.
We are interested in the irreducible representations of this finite group and therefore rep-resent the group elements by matrices, such that the generators fulfill the generating relations in eq. (4.6). Then an irreducible representation is realized, if there exists no change of basis that would simultaneously block-diagonalize all representation matrices. If such a change of basis does exists, the representation is reducible. It can then be written as a direct product of the representations that belong to its diagonal blocks.
For the hypercubic group there exist all in all twenty inequivalent irreducible representations.
Following [80] we label them by τkn with the superscript n giving the dimension of this rep-resentation and the optional subscript k counting inequivalent representations of the same dimension, if existent. Each of these irreducible representations is uniquely identified by the traces of its representation matrices τkn(G), called charactersχ of the representation:
χnk(G) =X
i
τkn(G)ii. (4.7)
For us the representation τ14 is of particular interest, because its 4×4-matrices describe how Lorentz vectors such as covariant derivatives transform under the group action. The related characters will allow to find the irreducibly transforming multiplets of three-quark operators.
The Spinorial Hypercubic Group
By now the transformation behavior of objects with integer spin under the symmetry group of the hypercubic lattice are known. These have to be generalized to half-integer spin since
4.2. CONSTRUCTION OF IRREDUCIBLE THREE-QUARK OPERATORS 55