Construction of Irreducible Three-Quark Operators

symmetry group was studied: the spinorial hypercubic group H(4). The overline expresses the fact that it is a double cover of H(4). This ensures that all the features of half-integer spin objects are realized that were absent for the non-spinorial counterpart, like the right phase factors for the lattice analogue of a full rotation. Therefore the defining relations of H(4) must be modified by twisting them with a set ofZ₂ factors, which results in a doubling of the group order:

I_i²=−1, I_iI_j =−I_jI_i tI₁ =I₁t, tI₂=I₄t, tI₃=I₂t, tI₄ =I₃t, γI1=−I₃, γI2=−I₂γ, γI4 =−I₄γ,

γ²=−1, t³=−1, (tγ)⁴ =−1. (4.8)

Again we are interested in the associated irreducible representations. Obviously representa-tions of H(4) are also representarepresenta-tions of H(4) and it can be shown that irreducible represen-tations of the former group are also irreducible with respect to the latter one. Hence the spinorial hypercubic group inherits all irreducible representations from H(4). Beyond that five further irreducible representations are found. These are “purely spinorial” and in order to distinguish them from the rest, we mark them with an underscore beneath their dimension, τ₁⁴, τ₂⁴, τ⁸, τ₁¹² and τ₂¹². (4.9) Here the four-dimensional representationτ₁⁴describes the transformation of four-spinors under the group action. Hence this representation will be of major importance, because it also defines the transformation behavior of the quark fields. And since we already know that derivatives transform according to τ₁⁴, we can deduce the transformation behavior of any three-quark operator, eq. (4.3), that is built out of three quark fields and a number of covariant derivatives. This allows the construction of irreducibly transforming three-quark operators in the next section.

4.2 Construction of Irreducible Three-Quark Operators

Let us now demonstrate, how the irreducibly transforming multiplets of three-quark operators can actually be derived from the representation matrices quoted above.

Knowing the representation matrices for spinors and Lorentz vectors one is in principle able to deduce the transformation of any three-quark operator (4.3) under any group element G ∈ H(4). To this end each spinor and Lorentz index is transformed separately with the representation matrices ofτ₁⁴ and τ₁⁴, respectively. This results in the G-transformed three-quark operator

O(j),G−transformed

=GijO⁽ⁱ⁾. (4.10)

However, we should keep in mind the large amount of independent operatorsO⁽ⁱ⁾. This sheer number would yield transformation matricesGji of rather unhandy dimension, and we want to avoid that. Therefore we will first study the transformation properties in the continuum groups SO4 and O4.

Table 4.2: Relation of quark fields with dotted and undotted indices to the Weyl representa-tion.

Weyl representation ψ1 ψ2 ψ3 ψ4

(un)dotted indices Φ⁰ Φ¹ Σ_˙0 Σ_˙1

chirality + + − −

This is reasonable, since the spinorial hypercubic group is embedded in the symmetry group of the Euclidean continuum O4. Hence one can be sure that irreducibly transforming operator multiplets of the latter one form a closed set with respect to the group action of H(4).

In other words: the H(4) representation matricesG_ij are block-diagonal with respect to the multiplets of three-quark operators that transform irreducibly under the continuum group.

This means that we only have to care about the associated lower-dimensional blocks ofG, once we know the multiplets of continuum-irreducible operators. Upon choosing appropriate linear combinations within these multiplets their blocks may decompose into even smaller blocks, which results in the desired H(4) irreducible representations. Using the symmetry group of the Euclidean continuum thus subdivides the search for H(4) irreducible three-quark operators in the whole operator space into two steps: First we construct the O4 irreducible multiplets, and then we look within them for H(4) irreducible multiplets of three-quark operators. This approach reduces the dimension of the problem considerably and finally fixes our choice for the coefficient tensorsT⁽ⁱ⁾ in eq. (4.3).

Irreducibility in SO₄ and O₄

Unless stated otherwise, we will focus on the leading-twist case from now on, i.e., twist = (mass dimension - spin) = 3. For notational convenience we furthermore write all quark fields with dotted and undotted indices in the chiral Weyl representation (cf. e.g., [82]).

Then a four-spinor naturally decomposes into two Weyl-spinors of definite chirality, compare Table 4.2, whose transformation properties are characterized by an SU(2) representation.

Analogously, we convert the covariant derivatives to an SU(2)×SU(2) representation by contracting them with the Pauli matricesσµ. Then the whole three-quark operator transforms as a direct product of SU(2) representations. This facilitates the construction of irreducible representations with respect to the space group, because there exists a homomorphism that links the irreducible representations of SU(2)×SU(2) to those of SO₄:

SU(2)×SU(2)'SO4. (4.11)

Due to this connection we can deduce irreducibly transforming three-quark operators for the latter group by constructing irreducible representations in SU(2)×SU(2), which is accom-plished by appropriately symmetrizing the SU(2) indices according to the corresponding stan-dard Young tableaux [83]. In leading-twist this enforces the independent total symmetrization of dotted and undotted indices.

Let us be more specific and exemplify the procedure by assuming that a particular com-bination of quark chiralities is given, i.e., the spinor indices are chosen to be either dotted

4.2. CONSTRUCTION OF IRREDUCIBLE THREE-QUARK OPERATORS 57 or undotted. Then an SO4 irreducibly transforming multiplet is constructed by the following two steps:

f_a˙g^bD_µ1. . . D_µnh^c → f_a˙g^b(Dσ)^d1_e˙

1. . .(Dσ)^dn_e˙

nh^c

→ f_{a˙g^{b(Dσ)^d1_e˙

1. . .(Dσ)^dn_e˙

n}h^c}, (4.12) where the brackets{. . .}denote the symmetrization of the indices on the same level. As each of the dotted and undotted indices can take the values zero or one, we immediately read off that this multiplet consists of (n+ 3)·(n+ 2) three-quark operators. Operators with other chirality combinations of the quark fields are treated in the same manner, so that the space of three-quark operators decomposes into subspaces of SO4 irreducible multiplets.

By reducing with respect to the double cover of the special orthogonal group in four dimensions, we have so far only taken four-dimensional rotations into account. The connec-tion to the full symmetry group of the Euclidean continuum O4 is established by reflection operations: letr represent some reflection in four dimensions, then [84]:

O₄= SO₄∪rSO₄. (4.13)

This relation holds also for the covering groups O₄ and SO₄. Consequently the O₄irreducible multiplets of three-quark operators can be constructed by combining any of the just deduced SO4 irreducible multiplets with its parity partner to a larger one. Thereby we have arrived at a basis of O₄ irreducible multiplets, with respect to which all representation matricesG_ij in eq. (4.10) are simultaneously block diagonal. In the following subsection this structure will be exploited for the further decomposition into the desired H(4) irreducible multiplets.

Irreducibility in H(4)

Before we explain how the actual decomposition works, it is interesting to have a look at the reduction from a more general point of view. One may ask the question, which of the H(4) irreducible representations may show up at all when reducing three-quark operators of the type (4.3). As stated repeatedly the covariant derivatives and quark fields transform according toτ₁⁴ andτ₁⁴, respectively. Therefore a three-quark operator transforms as a direct product of these representations:

τ₁⁴⊗τ₁⁴⊗τ₁⁴⊗ · · · ⊗τ₁⁴⊗τ₁⁴. (4.14) This product is reducible. Knowing the charactersχ^α for a given irreducible representation τ^α, it can be decomposed with the help of the identity

τ^α⊗τ^β =X

γ

c_γτ^γ, (4.15)

whereby the coefficients in the direct sum read cγ = 1

|H(4)|

X

G∈H(4)

χ^γ(G)^∗χ^α(G)χ^β(G). (4.16)

Here |H(4)|= 768 denotes the group order. Applying this formula iteratively, we derive the following content of H(4) irreducible multiplets for three-quark operators with zero to two derivatives (including higher twist):

zero derivatives: τ₁⁴⊗τ₁⁴⊗τ₁⁴ = 5τ₁⁴⊕τ⁸⊕3τ₁¹²,

one derivative: τ₁⁴⊗τ₁⁴⊗τ₁⁴⊗τ₁⁴ = 8τ₁⁴⊕4τ⁸⊕12τ₁¹²⊕4τ₂¹²,

two derivatives: τ₁⁴⊗τ₁⁴⊗τ₁⁴⊗τ₁⁴⊗τ₁⁴ = 20τ₁⁴⊕3τ₂⁴⊕18τ⁸⊕41τ₁¹²⊕23τ₂¹². (4.17) As expected, only spinorial representations show up after the reduction process.

We now proceed with the decomposition of the O4multiplets from the previous subsection.

To do so we must know the diagonal blocks assigned to these multiplets for all 768 trans-formation matrices G_ij in eq. (4.10) explicitly. For every group element this matrix block can be constructed in the following way. One transforms each quark field and derivative of a basis operator separately as explained above and then expresses the result again in terms of the basis operators. The coefficients involved are identified with the representation matrix elements Gij.

Once the representation matrices of the multiplets are known, we set up a projector to carry out the reduction. This projector contains also the characters of the desired H(4) irreducible representation. When it is applied to an O4 multiplet it projects out a usually smaller mul-tiplet that transforms irreducibly according to the desired representation (see, e.g., [85]):

P^α= dα

|H(4)|

X

G∈H(4)

χ^α(G)^∗G, (4.18)

where dα denotes the dimension of the irreducible representationτ^α to be projected out.

One finds that some O4 irreducible multiplets contain several equivalent H(4) irreducible representationsτ^α. Then the action ofP^α yields a set of three-quark operators that actually decomposes into smaller multiplets, which are closed under the group action on their own and irreducible. To separate these multiplets we introduce a second projector (see, e.g., [86]):

P˜_lk^α = dα

|H(4)|

X

G∈H(4)

(G^α_lk)^∗G. (4.19)

Here G^α_lk denotes the lk element of the representation matrix τ^α(G). Acting with the pro-jector ˜P₁₁^α on the set of operators in question results in m independent three-quark opera-tors, where m is the multiplicity of the representation τ^α. If we now apply the projectors {P˜_1j^α, j = 1, . . . , dα} to each of these m operators separately we generate m irreducible multiplets of three-quark operators. Thereby the required separation of the m equivalent irreducible multiplets is accomplished.

By now all irreducibly transforming three-quark operators of the spinorial hypercubic group are known. A more detailed review of this derivation can be found in [87].