Fierz transformations and renormalization schemes for four-quark operators

It has been shown that the choice of renormalization scheme is crucial for four-quark operators, in particular for neutral kaon mixing beyond the Standard Model. In the context of SMOM schemes, the choice of projector is not unique and is part of the definition of the renormalisation scheme. I present the non-diagonal Fierz relations which relate some of these projectors.


Introduction
A significant discrepancy has been observed for the four-quark operators matrix elements needed in the study of neutral kaon oscillations beyond the Standard Model.Beyond the quenched approximation, these quantities have been computed by RBC-UKQCD, [1,2], by ETM [3,4] and by SWME [5,6].The renormalisation factors have also been studied by Alpha [7].In collaboration with RBC-UKQCD, we have argued that the renormalisation procedure is responsible for this discrepancy: this was first reported in [8,9] and published in [10].We have shown that two SMOM schemes lead to very similar results after conversion to MS.However, for two of these quantities, these SMOM results are inconsistent with the ones obtained through the traditional RI-MOM intermediate scheme (with exceptional kinematics).We refer the reader to [11] for a review on the subject at this conference and to [12] for more details on the recent RBC-UKQCD results.
In Fig 1, we show that RI-MOM results are consistent within each other, regardless of the number of dynamical flavours, but significantly below the results obtained though RI-SMOM schemes.The latter are also compatible with SWME [5,6] where the renormalisation is performed at 1-loop in perturbation theory.In [13], we studied in details the RI-MOM and SMOM schemes, and pointed out potential issues with the RI-MOM procedure with exceptional kinematics.Using non-diagonal Fierz transformation, I show here that the SMOM schemes called (γ µ , γ µ ) and (/ q, / q) are mathematically different.These Fierz transformations presented here are general and can be used in different contexts.

∆S = Four quark operators
Within the Standard Model (SM), neutral kaon mixing involves the following colour unmixed fourquark operator where a and b are colour indices.The corresponding colour mixed operator reads One can show through a Fierz transformation that these operators are identical, ie Beyond the Standard Model, it is customary to introduce four extra operators.Using the same conventions, they can be written as two doublets, for example in the susy basis [14]: and

Fierz transformation
Denoting by Γ an arbitrary Dirac matrix, the four quark operators have the explicit form where Greek indices run in Dirac space.Swapping the Dirac indices β ↔ δ is equivalent to changing the color structure, from colour-mixed to colour-unmixed and vice-versa.However, according to Fierz theorem, one has where the indices i, j, k, l run from 1 to 16, and the 16 matrices Γ i form a basis of the four-by-four complex matrices.F is therefore a 16 4 tensor, whose entries are a priori unknown.
As a consequence, the colour mixed operators can be expressed in terms of linear combination of colour unmixed operators.We also note that Fierz transformation are properties of the Γ matrices, therefore the Fierz identities are exact on the lattice.However these identities only hold in fourdimension; in MS, these identities are modified by the presence of evanescent operators.

Choice of basis
It is possible to choose a basis in which all the operators are colour unmixed and where we omitted the colour indices.This choice can be found for example in [15].
We now separate the parity sectors, for example using the standard notation We can derive the well-known Fierz identities, in Eucledian space time we find see for example [16].We note the mixing pattern expected from chiral symmetry is respected, ie VV + AA renormalises multiplicatively, VV − AA and S S − PP mix together, and so do S S + PP and T T .However these identities are a restriction to the diagonal case, ie i = j in Eq.( 7).Other Fierz relations can also be found in [17].

Choice of projectors
Following the Rome-Southampton method, the renormalisation of a four-quark operators O requires the projection of an amputated green function where T is the tree level value and P is a projector in Dirac and colour space In the operator mixing case P and Λ O are vectors, P [Λ O ] and T are matrices.
As an example, let us consider the standard model operator VV + AA.A natural choice for P is We have defined the projector with the same Dirac-colour structure as the operator.Such a projector is called a γ µ -projector ≡ P γ µ .Note that we can also define However, if a SMOM scheme is implemented, there is a non zero momentum transfer q = p 2 − p 1 .Therefore we can also define a so-called / q-projector [18] The choice of Projector is part of the definition of the scheme, they lead to (a priori) different non-perturbative Z factors and have different MS conversion factors.(After conversion to MS, the Z factors should agree up to truncation error in the perturbative expansion and to discretisation artefacts.).
We turn now to the case of the (8, 8) operators (Q 2,3 or O 4,5 ).We define / T 5 / T 5 = q µ q ν σ µρ γ 5 × σ νρ γ 5 , and derive the following Fierz matrix We recognise the γ µ -projectors in the top-left corner, and we check that the / q are not linear combinations of the γ µ -projectors.Therefore, the γ µ and / q-projectors defined above (and in [13]) lead to independent renormalisation schemes.