AJB-22-1

Simple Rules for Evanescent Operators in One-Loop Basis Transformations

Jason Aebischera, Andrzej J. Burasb and Jacky Kumarb

aPhysik-Institut, Universität Zürich, CH-8057 Zürich, Switzerland
bTUM Institute for Advanced Study, Lichtenbergstr. 2a, D-85747 Garching,
Germany

###### Abstract

Basis transformations often involve Fierz and other relations which are only
valid in $D=4$ dimensions. In general $D$ space-time dimensions however,
evanescent operators have to be introduced, in order to preserve such
identities. Such evanescent operators contribute to one-loop basis
transformations as well as to two-loop renormalization group running. We
present a simple procedure on how to systematically change basis at the one-
loop level by obtaining shifts due to evanescent operators. As an example we
apply this method to derive the one-loop basis transformation from the BMU
(Buras, Misiak and Urban) basis useful for NLO QCD calculations, to the JMS
(Jenkins, Manohar and Stoffer) basis used in the matching to the SMEFT.

###### Contents

  1. 1 Introduction
  2. 2 Procedure
    1. 2.1 Tree-level transformation
    2. 2.2 One-loop transformation
    3. 2.3 How to use this procedure
  3. 3 BMU to JMS translation at one-loop
    1. 3.1 Basic method
    2. 3.2 Transformation matrices at one-loop
  4. 4 Evanescent operators
    1. 4.1 Definition of EV operators in the VLL sector
    2. 4.2 Calculation of the EV operators in the VLL sectors
    3. 4.3 Definition of EV operators in the SRR sector
    4. 4.4 Calculation of EV operators in the SRR sector
    5. 4.5 Definition of EV operators in the VLR sector
  5. 5 Conclusions
  6. A $\Delta F=1$ BMU basis for $N_{f}=5$
  7. B Fierz identities
  8. C Master Formulae for One-loop Operator Insertions
    1. C.1 Open penguin insertion
    2. C.2 Closed penguin insertion
    3. C.3 Special Cases

##  1 Introduction

In recent years a lot of progress has been made concerning Next-to-Leading
Order (NLO) analyses, which involve one-loop matching calculations as well as
solving two-loop Renormalization Group Equations (RGEs) for the Wilson
coefficients of Effective Field Theories. For an up to date review see [1].
For instance, concerning the Standard Model Effective Theory (SMEFT), the full
matching from the SMEFT onto the Weak Effective Theory (WET) valid below the
electroweak (EW) scale is known at tree-level [2] and since recently also at
the one-loop level [3, 4]. Furthermore, the one-loop RGEs in the SMEFT [5, 6,
7] and in the WET [8, 9] are known.

In the process of performing a NLO analysis, it is often necessary to perform
one-loop transformations between different operator bases, since for instance
anomalous dimension matrices (ADMs) are known only in a particular basis,
whereas the matching conditions are given in a different one. This is for
example the case for the WET, where the two-loop ADMs are known in the BMU
(Buras, Misiak and Urban) basis [10] as elaborated on recently in [11]. On the
other hand the one-loop matching from SMEFT onto WET is given in the JMS
(Jenkins, Manohar and Stoffer) basis defined in [2].111We follow here the WCxf
convention defined in [12].

Generally, in order to translate the results from one basis to another one at
NLO, one-loop basis transformations have to be taken into account. In this
respect particular care has to be taken, if the tree-level (Leading Order
(LO)) transformations involve Dirac space identities, which are only valid in
four space-time dimensions. Examples include Fierz identities or identities
involving gamma matrices. When using dimensional regularization, where the
divergent loop integrals are continued to $D$ dimensions, such identities can
not be used directly, but need to be generalized by introducing evanescent
(EV) operators [10]. Such evanescent operators vanish in four dimensions to
conserve the original identities but are non-zero in $D$ dimensions. They are
therefore formally speaking proportional to $\epsilon=(4-D)/2$, which implies
that they give non-zero contributions when inserted into divergent loop
diagrams. These are exactly the contributions which enter basis
transformations at the one-loop order. In this article we discuss a simple
procedure on how to obtain these contributions by computing one-loop
corrections resulting from the presence of the evanescent operators.222A more
formal procedure on how to perform NLO basis changes can be found in [10]. To
this end the evanescent operators are simply defined as the difference between
operators and their transformed versions using $D=4$ identities. The resulting
contributions will manifest themselves in shifts in the corresponding Wilson
coefficients of the initial operators.

Having a simple algorithm at hand to perform NLO basis changes is important
when performing one-loop matching calculations or in the computation of two-
loop ADMs. In this work we explain the underlying formal framework and provide
such an algorithm, based on Greek projections [13], which facilitates the
aforementioned calculations. In this manner this procedure is an important
ingredient in the pursuit of a complete NLO SMEFT analysis.

The rest of the article is organized as follows: In Section 2 we outline the
general procedure on how to compute one-loop basis transformations between two
operator bases. In Section 3 we show an explicit example by performing a NLO
change from the BMU to the JMS basis. In Section 4 we define and calculate the
EV operators which in turn gives us the transformation matrix between BMU and
JMS bases at the one-loop level. Finally we conclude in Section 5. Additional
material used in the calculations is collected in the appendices.

##  2 Procedure

In this section we discuss the full NLO basis transformation between general
operator bases. However, in order to make the subsequent sections more
transparent we will dub the two operator bases JMS (see Table. 1 for the list
of operators considered in this work) and BMU. We start with the simple LO
basis transformation which will also set the notation used. Then, in the
second subsection we discuss the issue of evanescent operators which become
relevant when performing basis changes at the one-loop level.

$(\bar{L}L)(\bar{L}L)$ | $(\bar{R}R)(\bar{R}R)$
---|---
$[O_{dd}^{V,LL}]_{prst}$ | $(\bar{d}_{L}^{p}\gamma_{\mu}d_{L}^{r})(\bar{d}_{L}^{s}\gamma^{\mu}d_{L}^{t})$ | $[O_{dd}^{V,RR}]_{prst}$ | $(\bar{d}_{R}^{p}\gamma_{\mu}d_{R}^{r})(\bar{d}_{R}^{s}\gamma^{\mu}d_{R}^{t})$
$[O_{ud}^{V1,LL}]_{prst}$ | $(\bar{u}_{L}^{p}\gamma_{\mu}u_{L}^{r})(\bar{d}_{L}^{s}\gamma^{\mu}d_{L}^{t})$ | $[O_{ud}^{V1,RR}]_{prst}$ | $(\bar{u}_{R}^{p}\gamma_{\mu}u_{R}^{r})(\bar{d}_{R}^{s}\gamma^{\mu}d_{R}^{t})$
$[O_{ud}^{V8,LL}]_{prst}$ | $(\bar{u}_{L}^{p}\gamma_{\mu}T^{A}u_{L}^{r})(\bar{d}_{L}^{s}\gamma^{\mu}T^{A}d_{L}^{t})$ | $[O_{ud}^{V8,RR}]_{prst}$ | $(\bar{u}_{R}^{p}\gamma_{\mu}T^{A}u_{R}^{r})(\bar{d}_{R}^{s}\gamma^{\mu}T^{A}d_{R}^{t})$
$(\bar{L}L)(\bar{R}R)$ |  $(\bar{L}R)(\bar{L}R)$\+ h.c.
$[O_{dd}^{V1,LR}]_{prst}$ | $(\bar{d}_{L}^{p}\gamma_{\mu}d_{L}^{r})(\bar{d}_{R}^{s}\gamma^{\mu}d_{R}^{t})$ | $[O_{dd}^{S1,RR}]_{prst}$ | $(\bar{d}_{L}^{p}d_{R}^{r})(\bar{d}_{L}^{s}d_{R}^{t})$
$[O_{dd}^{V8,LR}]_{prst}$ | $(\bar{d}_{L}^{p}\gamma_{\mu}T^{A}d_{L}^{r})(\bar{d}_{R}^{s}\gamma^{\mu}T^{A}d_{R}^{t})$ | $[O_{dd}^{S8,RR}]_{prst}$ | $(\bar{d}_{L}^{p}T^{A}d_{R}^{r})(\bar{d}_{L}^{s}T^{A}d_{R}^{t})$
$[O_{ud}^{V1,LR}]_{prst}$ | $(\bar{u}_{L}^{p}\gamma_{\mu}u_{L}^{r})(\bar{d}_{R}^{s}\gamma^{\mu}d_{R}^{t})$ | $[O_{ud}^{S1,RR}]_{prst}$ | $(\bar{u}_{L}^{p}u_{R}^{r})(\bar{d}_{L}^{s}d_{R}^{t})$
$[O_{ud}^{V8,LR}]_{prst}$ | $(\bar{u}_{L}^{p}\gamma_{\mu}T^{A}u_{L}^{r})(\bar{d}_{R}^{s}\gamma^{\mu}T^{A}d_{R}^{t})$ | $[O_{ud}^{S8,RR}]_{prst}$ | $(\bar{u}_{L}^{p}T^{A}u_{R}^{r})(\bar{d}_{L}^{s}T^{A}d_{R}^{t})$
$[O_{du}^{V1,LR}]_{prst}$ | $(\bar{d}_{L}^{p}\gamma_{\mu}d_{L}^{r})(\bar{u}_{R}^{s}\gamma^{\mu}u_{R}^{t})$ | $[O_{uddu}^{S1,RR}]_{prst}$ | $(\bar{u}_{L}^{p}d_{R}^{r})(\bar{d}_{L}^{s}u_{R}^{t})$
$[O_{du}^{V8,LR}]_{prst}$ | $(\bar{d}_{L}^{p}\gamma_{\mu}T^{A}d_{L}^{r})(\bar{u}_{R}^{s}\gamma^{\mu}T^{A}u_{R}^{t})$ | $[O_{uddu}^{S8,RR}]_{prst}$ | $(\bar{u}_{L}^{p}T^{A}d_{R}^{r})(\bar{d}_{L}^{s}T^{A}u_{R}^{t})$
$[O_{uddu}^{V1,LR}]_{prst}$ |  $(\bar{u}_{L}^{p}\gamma_{\mu}d_{L}^{r})(\bar{d}_{R}^{s}\gamma^{\mu}u_{R}^{t})$ \+ h.c. |  |
$[O_{uddu}^{V8,LR}]_{prst}$ |  $(\bar{u}_{L}^{p}\gamma_{\mu}T^{A}d_{L}^{r})(\bar{d}_{R}^{s}\gamma^{\mu}T^{A}u_{R}^{t})$ \+ h.c. |  |
Table 1: Non-leptonic $\Delta F=1$ operators (baryon and lepton number
conserving) in the JMS basis [2]. Note that $Q_{uddu}^{V1,LR}$ and
$Q_{uddu}^{V8,LR}$ have Hermitian conjugates. The same holds for the operators
$(\overline{L}R)(\overline{L}R)$. This choice of basis eliminates all
operators with Dirac structures $\sigma^{\mu\nu}$. The class of operators
$(\overline{L}R)(\overline{R}L)+\text{h.c.}$ does not contain non-leptonic
operators, but only semi-leptonic ones.

###  2.1 Tree-level transformation

Let us consider the two bases

$\vec{\mathcal{O}}_{\rm
BMU}=\\{Q_{1},Q_{2},...,Q_{N}\\},\qquad\vec{\mathcal{O}}_{\rm
JMS}=\\{O_{1},O_{2},...,O_{N}\\},$ (1)

containing $N$ operators each. At tree-level each operator is given by a
linear combination of operators from the other basis:

$\vec{\mathcal{O}}_{\rm JMS}=\hat{R}^{(0)}\,\vec{\mathcal{O}}_{\rm BMU},$ (2)

where the $N\times N$ matrix $\hat{R}^{(0)}$ denotes the linear transformation
between the two bases. The superindex ’$(0)$’ denotes the tree-level
transformation in anticipation of the one-loop transformation discussed in the
next subsection. The matrix $\hat{R}^{(0)}$ is obtained by applying identities
such as Fierz relations and gamma matrix identities to the operators on the
LHS of eq. (2). It is independent of the renormalization scale and only
contains numerical factors.333This is true up to possible normalization
factors, which can contain coupling constants or other parameters that depend
on the renormalization scale.

As seen in eq. (2) the BMU operator basis is transformed via $\hat{R}^{(0)}$
into the JMS basis. This transformation is useful because the hadronic matrix
elements of operators are usually calculated in the BMU basis and this
transformation allows to obtain them in the JMS basis. However, the Wilson
coefficients are nowadays calculated in the JMS basis, since the SMEFT
matching results are only available in that particular basis. Therefore for
Wilson coefficients the transformation from JMS to BMU is more useful, which
reads

$\vec{\mathcal{C}}^{(0)}_{\rm
BMU}=(\hat{R}^{(0)})^{T}\,\vec{\mathcal{C}}^{(0)}_{\rm JMS}\,,$ (3)

that is $(\hat{R}^{(0)})^{T}$ is involved.

Since $\hat{R}^{(0)}$ is invertible by definition, one can easily express the
BMU operators in terms of JMS ones and the JMS Wilson coefficients in terms of
BMU ones. For our purposes however, the transformations in eqs. (2) and (3)
are more convenient.

It should be stressed that although the procedure outlined below is general
and can be used for any basis transformation the explicit values of various
coefficients are given in our paper in the NDR-$\overline{\text{MS}}$ scheme
as defined in [14] with evanescent operators entering two-loop calculations
defined by the so-called Greek method. The details in the context of WET and
SMEFT are discussed in Appendix E of [15].

###  2.2 One-loop transformation

At the one-loop level special care has to be taken when identities have been
used in the LO transformation, that are only valid in $D=4$ space-time
dimensions. Therefore, for all the lines in $R^{(0)}$ which were obtained
using $D=4$ relations, evanescent operators have to be introduced to
generalize these identities. In the case of the BMU$\rightarrow$JMS
transformation we proceed as follows:

Step 1:

We perform a Fierz transformation on every operator in the BMU basis $Q_{i}$
and denote the result of this transformation by $\widetilde{Q}_{i}$ given
generally by

${\widetilde{Q}_{i}=\sum_{k}\omega_{ik}Q_{k}\,,}$ (4)

with operators $Q_{k}$ belonging to the BMU basis and coefficients
$\omega_{ik}$ determined through the Fierz identities collected in the
Appendix B. It should be emphasized that the relation above and analogous
relations below should be interpreted as effective contributions from this
operator in one loop diagrams. This means that the calculations performed in
two bases, in our case BMU and JMS, will give the same results for physical
observables when this transformation is taken into account at all stages,
therefore when calculating also anomalous dimensions. This relations and
analogous relations involving evanescent operators can also be used for the
calculations of two-loop anomalous dimensions of operators because there the
evanescent operators contributing to two-loop anomalous dimensions appear in
one-loop sub-diagrams [14]. In order to go to the next order in perturbation
theory the shifts should include corrections of order
$\mathcal{O}\left(\alpha_{s}^{2}\right)$ that can be obtained through
insertions of the operators into two-loop diagrams.

Step 2:

We insert $Q_{i}$ and $\widetilde{Q}_{i}$ defined by (4) into current-current
and QCD penguin diagrams of Figs. 1 and 2, respectively. Due to the presence
of evanescent operators which are simply defined by

$Q_{i}=\widetilde{Q}_{i}+\text{EV}_{i}\,,$ (5)

the insertion of $Q_{i}$ and $\widetilde{Q}_{i}$ into one-loop diagrams will
generally differ at $\mathcal{O}(\alpha_{s})$, leading to the result

$Q_{i}=\widetilde{Q}_{i}+\frac{\alpha_{s}}{4\pi}\sum_{r}{\tilde{\omega}_{ir}}Q_{r},$
(6)

where the operators in the sum can again be written in the BMU basis but are
generally different from the ones in the definition of $\widetilde{Q}_{i}$ in
(4) and also $\omega_{ik}\not=\tilde{\omega}_{ir}$. Note that this can easily
be done at the one-loop level because in transforming the operators entering
these corrections $D=4$ Fierz identities can be used as will be explained
below. Expressing the shift in terms of BMU operators allows to use these
results for transformations of the BMU basis to any basis, not just the JMS
one.

To compute the $\alpha_{s}$-corrections resulting from the evanescent operator
$\text{EV}_{i}$ in (5), one simply inserts the difference
$Q_{i}-\widetilde{Q}_{i}$ into the relevant one-loop diagrams. Since the
evanescent operator is formally $\mathcal{O}(\epsilon)$, only the divergent
pieces of the loop integrals will contribute and consequently the finite
pieces can be discarded in the calculation.

The relation between operators in JMS basis and the Fierz transformed
operators in BMU are further discussed in Sec.3.2. In particular the tree-
level relations are given in eqs. (46)-(49), and the one-loop relations in
eqs. (50)-(52).

Step 3:

Having the relations in (6) and inspecting which Fierz transformations had to
be performed to find the LO matrix $\hat{R}^{(0)}$ we can generalize the basis
change matrix $\hat{R}^{(0)}$ to the one-loop order, as follows:444In the case
JMS$\leftrightarrow$BMU we will focus on QCD corrections. The relation in eq.
(7) can easily be generalized to include other one-loop corrections.

$\vec{\mathcal{O}}_{\rm JMS}=\hat{R}\,\vec{\mathcal{O}}_{\rm
BMU}\,,\qquad\hat{R}=\hat{R}^{(0)}+\frac{\alpha_{s}}{4\pi}\hat{R}^{(1)}\,,$
(7)

where $\hat{R}^{(1)}$ is the one-loop basis transformation matrix resulting
exclusively from the evanescent operators.

When computing the one-loop corrections to the difference
$Q_{i}-\widetilde{Q}_{i}$, new Dirac structures can appear, which are not
present in the original basis, in our case the BMU basis. To reduce these
structures to the ones in $\vec{\mathcal{O}}_{\rm BMU}$ it is essential to use
the same projections as in the calculations of two-loop anomalous dimensions
of the operators. Only then are the evanescent operators entering the two-loop
calculations the same as the ones used in the one-loop matching between WET
and SMEFT and in particular they correspond to the ones in the basis change.
This prescription guarantees that the renormalization scheme dependence of
two-loop matrix elements can be canceled by the one present in one-loop
contributions so that the physical amplitudes are renormalization scheme
independent.

These issues have been discussed in the context of the NLO QCD calculations of
Wilson coefficients at length in [14, 16, 17] and the summary can be found in
Section 5.2.9 of [1]. There the so-called Greek projection [13], properly
generalized to include evanescent operators in [14], has been discussed in
detail. As pointed out in [17] this is not the only way to include evanescent
operators but in fact the simplest one. It has been used in all two-loop
calculations performed by the second author and will be used in the following
but this time in the context of basis change. It should be kept in mind that
this procedure defines the evanescent operators in the
NDR-$\overline{\text{MS}}$ scheme combined with the Greek projections as used
in [14]. While giving the same results it is much simpler than the formal
method presented in [10] and used recently in [11].

The NLO transformation in terms of the Wilson coefficients is given as
follows.

$\vec{\mathcal{C}}_{\rm BMU}=\hat{R}^{T}\vec{\mathcal{C}}_{\rm JMS}.$ (8)

Writing

$\vec{\mathcal{C}}_{\rm JMS}=\vec{\mathcal{C}}^{(0)}_{\rm
JMS}+\frac{\alpha_{s}}{4\pi}\vec{\mathcal{C}}^{(1)}_{\rm
JMS}\,,\qquad\vec{\mathcal{C}}_{\rm BMU}=\vec{\mathcal{C}}^{(0)}_{\rm
BMU}+\frac{\alpha_{s}}{4\pi}\vec{\mathcal{C}}^{(1)}_{\rm BMU}\,,$ (9)

the transformation for Wilson coefficients reads

$\vec{\mathcal{C}}^{(0)}_{\rm
BMU}=(\hat{R}^{(0)})^{T}\vec{\mathcal{C}}^{(0)}_{\rm
JMS}\,,\qquad\vec{\mathcal{C}}^{(1)}_{\rm
BMU}=(\hat{R}^{(0)})^{T}\vec{\mathcal{C}}^{(1)}_{\rm
JMS}+(\hat{R}^{(1)})^{T}\vec{\mathcal{C}}^{(0)}_{\rm JMS}\,.$ (10)

A few remarks concerning the generality of our results are in order: The
procedure laid out in this article is not limited to the BMU and JMS basis,
but is valid for any pair of operator bases, that are related via Fierz
transformations. The full set of relevant evanescent operators is generated by
applying Greek identities, or any other identities to reduce the resulting
Dirac structures from the one-loop calculation. This choice of Dirac
reduction, together with the chosen finite counterterms and the treatment of
$\gamma_{5}$ fixes the renormalization scheme [17]. In this article we have
chosen the NDR-$\overline{\text{MS}}$ scheme in combination with the Greek
identities. Choosing a different renormalization scheme would change the
entries of the rotation matrix $R^{(1)}$ in eq. (7), but is related via a
trivial change of scheme to our results [18, 19]. Furthermore, the procedure
is independent of possible group theory relations between the two bases, since
these are still valid in $D\neq 4$ space-time dimensions, and therefore do not
obtain any one-loop shifts from evanescent structures. Finally, we note that
the operator shifts presented in this paper can be interpreted as one-loop
corrections to the original Fierz identities. This has been shown in two
recent publications [20, 21], in which the shifts for all possible four-fermi
operators together with the contributions from dipole operators have been
taken into account. A similar procedure in the SMEFT has been employed in
[22].

###  2.3 How to use this procedure

Having the results in (6) to be presented in the next section, our goal will
be to find the matrix $\hat{R}$. To this end comparing the BMU and JMS bases
one has to find those operators or groups of them for which a Fierz
transformation on operators in the BMU basis has to be performed in order to
obtain the operators in the JMS basis with order $\alpha_{s}$ corrections
taken into account.

Generally the matrix $\hat{R}^{(0)}$ will have a block structure so that
operators in a given block can be separately considered from other blocks. In
this context the following three cases arise.

  * •

If no Fierz transformations are required in a given block the corresponding
matrix $\hat{R}^{(1)}$ will vanish and tree-level results will be valid also
at one-loop. One can use the corresponding block in $\hat{R}^{(0)}$ in that
case.

  * •

If Fierz transformations in a given block are required but the contributions
of evanescent operators will vanish the corresponding block in the tree-level
matrix $\hat{R}^{(0)}$ will again represent the corresponding block in the
full $\hat{R}$.

  * •

Finally, in certain blocks the necessity of performing Fierz transformations
will introduce evanescent operators which will contribute to $\hat{R}^{(1)}$.

In the next section we will present the three step procedure outlined in this
section in explicit terms.

##  3 BMU to JMS translation at one-loop

###  3.1 Basic method

As outlined above the transformation of the BMU basis to the JMS basis
requires Fierz transformations on some of the BMU operators. This generates
then additional contributions to the one-loop matching performed within the
JMS basis. In order to find these one loop contributions one has to insert the
difference $Q_{i}-\widetilde{Q}_{i}$ into current-current and penguin diagrams
of Figs. 1 and 2, respectively. The details of the calculation for the 1-loop
insertions are given in Appendix C.

(a)

(b)

(c)

Figure 1:  The QCD current-current insertions at one-loop.

(a)

(b)

Figure 2:  The QCD-penguin insertions with open–type [left] and closed–type
[right] fermion loops.

With all BMU operators listed in Appendix A it turns out that

  * •

For $Q_{k}$ with $k=1-18$ these additional contributions come only from
penguin insertions and moreover only for a few among these operators listed
below.

  * •

Fierz transformations on $Q_{1}^{\text{SLR},Q}$ and $Q_{2}^{\text{SLR},Q}$ do
not generate any evanescent contributions and consequently in this case $D=4$
identities can be used.

  * •

For $Q_{k}^{\text{SRR},Q}$ with $k=1-4$ and $Q_{l}^{\text{SRR},D}$ with
$l=1,2$ the contributions come only from current-current operators and involve
all operators considered. However, all these operators do not contribute to
$K$ and $B$ decays being forbidden within SMEFT. For completeness we list
these contributions below because they could be useful for charm physics.

In the case of the SM operators $Q_{k}$ with $k=1-10$ all the contributions
from Fierz transformations for LL (left-left) operators can be obtained from
two properties:

$Q_{1}=\widetilde{Q}_{1},\qquad
Q_{2}=\widetilde{Q}_{2}+\frac{1}{3}\frac{\alpha_{s}}{4\pi}P,$ (11)

with

$P=Q_{4}+Q_{6}-\frac{1}{3}(Q_{3}+Q_{5}).$ (12)

We find

$Q_{3}=\widetilde{Q}_{3}+\frac{2}{3}\frac{\alpha_{s}}{4\pi}P,\qquad
Q_{4}=\widetilde{Q}_{4}-\frac{N_{f}}{3}\frac{\alpha_{s}}{4\pi}P,\qquad
Q_{5}=\widetilde{Q}_{5},\qquad Q_{6}=\widetilde{Q}_{6}\,,$ (13)
$Q_{7}=\widetilde{Q}_{7},\qquad Q_{8}=\widetilde{Q}_{8},\qquad
Q_{9}=\widetilde{Q}_{9}-\frac{1}{3}\frac{\alpha_{s}}{4\pi}P,\qquad
Q_{10}=\widetilde{Q}_{10}-\frac{1}{3}(N_{u}-\frac{N_{d}}{2})\frac{\alpha_{s}}{4\pi}P.$
(14)

We observe that the Fierz transformations on the VLR operators $Q_{k}$ with
$k=5-8$ do not bring any contributions from evanescent operators.

In the case of the NP operators $Q_{k}$ with $k=11-18$ only the Fierz
transformation on $Q_{11}$ brings a contribution from evanescent operators so
that

$Q_{11}=\widetilde{Q}_{11}+\frac{2}{3}\frac{\alpha_{s}}{4\pi}P,\qquad
Q_{k}=\widetilde{Q}_{k},\quad k=12-18\,.$ (15)

The corresponding results for $Q_{1,2,3,4}^{\text{SRR},D}$ with $D=d_{i}$ or
$d_{j}$ operators can be obtained by using the results of [23], in particular
the results in equations (29)-(32) of that paper. In this case the
$\widetilde{Q}_{k}^{\text{SRR},D}$ operators with $k=1-4$ are given as
follows:

$\displaystyle\widetilde{Q}_{1}^{\text{SRR},D}$
$\displaystyle=-\frac{1}{2}Q_{2}^{\text{SRR},D}+\frac{1}{8}Q_{4}^{\text{SRR},D},$
$\displaystyle\widetilde{Q}_{2}^{\text{SRR},D}$
$\displaystyle=-\frac{1}{2}Q_{1}^{\text{SRR},D}+\frac{1}{8}Q_{3}^{\text{SRR},D},$
(16) $\displaystyle\widetilde{Q}_{3}^{\text{SRR},D}$
$\displaystyle=6Q_{2}^{\text{SRR},D}+\frac{1}{2}Q_{4}^{\text{SRR},D},$
$\displaystyle\widetilde{Q}_{4}^{\text{SRR},D}$
$\displaystyle=6Q_{1}^{\text{SRR},D}+\frac{1}{2}Q_{3}^{\text{SRR},D}.$ (17)

As this time only current-current diagrams are involved the flavour structure
relative to the one considered in [23] does not matter and the full
calculation of the matrix elements of $\widetilde{Q}_{i}$ operators can be
readily performed in no time using results of [23]. The shifts caused by
evanescent operators involve four operators but those with $k=1,3$ can be
eliminated555Equivalently one could also eliminate any other two operators but
here we follow the conventions used in Ref.[11]. using

$\displaystyle Q_{3}^{\text{SRR},D}$
$\displaystyle=6\,Q_{2}^{\text{SRR},D}+\frac{1}{2}\,Q_{4}^{\text{SRR},D}+\text{Fierz
ev.}$ (18) $\displaystyle Q_{1}^{\text{SRR},D}$
$\displaystyle=-\frac{1}{2}\,Q_{2}^{\text{SRR},D}+\frac{1}{8}\,Q_{4}^{\text{SRR},D}+\text{Fierz
ev.}$ (19)

so that the shifts depend only on $Q_{2,4}^{\text{SRR},D}$. Note that in the
BMU basis $Q_{2,4}^{\text{SRR},D}$ are denoted as $Q_{1,2}^{\text{SRR},i}$ or
$Q_{1,2}^{\text{SRR},j}$ in Ref.[11], see Appendix A for definitions.

We find then

$\displaystyle Q_{1}^{\text{SRR},D}$
$\displaystyle=\widetilde{Q}_{1}^{\text{SRR,D}}+\frac{\alpha_{s}}{4\pi}\sum_{k=2,4}A_{k}Q_{k}^{\text{SRR,D}},$
(20) $\displaystyle Q_{2}^{\text{SRR},D}$
$\displaystyle=\widetilde{Q}_{2}^{\text{SRR,D}}+\frac{\alpha_{s}}{4\pi}\sum_{k=2,4}B_{k}Q_{k}^{\text{SRR,D}},$
(21) $\displaystyle Q_{3}^{\text{SRR},D}$
$\displaystyle=\widetilde{Q}_{3}^{\text{SRR,D}}+\frac{\alpha_{s}}{4\pi}\sum_{k=2,4}C_{k}Q_{k}^{\text{SRR,D}},$
(22) $\displaystyle Q_{4}^{\text{SRR},D}$
$\displaystyle=\widetilde{Q}_{4}^{\text{SRR,D}}+\frac{\alpha_{s}}{4\pi}\sum_{k=2,4}D_{k}Q_{k}^{\text{SRR,D}},$
(23)

with the coefficients $A_{k},B_{k},C_{k},D_{k}$ given as follows666$N_{c}$ is
the number of colours with $N_{c}=3$ in the final results.

$\displaystyle A_{2}$
$\displaystyle=\frac{1}{2}+\frac{5}{N_{c}}-\frac{7N_{c}}{4}=-\frac{37}{12},$
$\displaystyle A_{4}$
$\displaystyle=-\frac{1}{2}+\frac{1}{4N_{c}}-\frac{N_{c}}{16}=-\frac{29}{48}\,.$
(24)

$\displaystyle B_{2}$
$\displaystyle=-\frac{17}{4}-\frac{1}{N_{c}}=-\frac{55}{12},$ $\displaystyle
B_{4}$
$\displaystyle=-\frac{3}{16}+\frac{3}{4N_{c}}-\frac{N_{c}}{8}=-\frac{5}{16}\,.$
(25)

$\displaystyle C_{2}$ $\displaystyle=36+\frac{28}{N_{c}}-7N_{c}=\frac{73}{3},$
$\displaystyle C_{4}$
$\displaystyle=-\frac{1}{2}-\frac{5}{N_{c}}+\frac{3N_{c}}{4}=\frac{1}{12}\,.$
(26)

$\displaystyle D_{2}$
$\displaystyle=-21-\frac{44}{N_{c}}+14N_{c}=\frac{19}{3},$ $\displaystyle
D_{4}$ $\displaystyle=\frac{13}{4}+\frac{1}{N_{c}}=\frac{43}{12}\,.$ (27)

The same procedure can be applied to $Q_{1,2,3,4}^{\text{SRR},Q}$ with
$Q=u_{k},d_{k}\neq d_{i},d_{j}$ defined in eq. (100). The rules for the shifts
read

$\displaystyle Q_{1}^{\text{SRR},Q}$
$\displaystyle=\widetilde{Q}_{1}^{\text{SRR,Q}}+\frac{\alpha_{s}}{4\pi}\sum_{k=1,2,3,4}a_{k}Q_{k}^{\text{SRR,Q}}\,,$
(28) $\displaystyle Q_{2}^{\text{SRR},Q}$
$\displaystyle=\widetilde{Q}_{2}^{\text{SRR,Q}}+\frac{\alpha_{s}}{4\pi}\sum_{k=1,2,3,4}b_{k}Q_{k}^{\text{SRR,Q}}\,,$
(29) $\displaystyle Q_{3}^{\text{SRR},Q}$
$\displaystyle=\widetilde{Q}_{3}^{\text{SRR,Q}}+\frac{\alpha_{s}}{4\pi}\sum_{k=1,2,3,4}c_{k}Q_{k}^{\text{SRR,Q}}\,,$
(30) $\displaystyle Q_{4}^{\text{SRR},Q}$
$\displaystyle=\widetilde{Q}_{4}^{\text{SRR,Q}}+\frac{\alpha_{s}}{4\pi}\sum_{k=1,2,3,4}d_{k}Q_{k}^{\text{SRR,Q}}\,.$
(31)

Here the flavour structure of the tilde operators
$\widetilde{Q}_{1,2,3,4}^{\text{SRR},Q}$ (see (16),(17) for the definition
with $D$ replaced by $Q$) is $(\bar{d}_{j}\Gamma Q)(\bar{Q}\Gamma d_{i})$ and
the BMU operators $Q_{1,2,3,4}^{\text{SRR},Q}$ have the form
$(\bar{d}_{j}\Gamma d_{i})(\bar{Q}\Gamma Q)$.

The coefficients $a_{k},b_{k},c_{k},d_{k}$ are given as follows

$\displaystyle a_{1}$
$\displaystyle=\frac{N_{c}}{2}-\frac{1}{N_{c}}=\frac{7}{6},$ $\displaystyle
a_{2}$ $\displaystyle=\frac{1}{2},$ (32) $\displaystyle a_{3}$
$\displaystyle=-\frac{N_{c}}{4}+\frac{3}{4N_{c}}=-\frac{1}{2},$ $\displaystyle
a_{4}$ $\displaystyle=-\frac{1}{2}\,.$ (33)

$\displaystyle b_{1}$ $\displaystyle=1,$ $\displaystyle b_{2}$
$\displaystyle=-\frac{1}{N_{c}}=-\frac{1}{3},$ (34) $\displaystyle b_{3}$
$\displaystyle=-\frac{5}{8},$ $\displaystyle b_{4}$
$\displaystyle=-\frac{N_{c}}{8}+\frac{3}{4N_{c}}=-\frac{1}{8}.$ (35)

$\displaystyle c_{1}$ $\displaystyle={8}N_{c}-\frac{44}{N_{c}}=\frac{28}{3},$
$\displaystyle c_{2}$ $\displaystyle=36,$ (36) $\displaystyle c_{3}$
$\displaystyle=-\frac{N_{c}}{2}+\frac{1}{N_{c}}=-\frac{7}{6},$ $\displaystyle
c_{4}$ $\displaystyle=-\frac{1}{2}.$ (37)

$\displaystyle d_{1}$ $\displaystyle=30,$ $\displaystyle d_{2}$
$\displaystyle=14{N_{c}}-\frac{44}{N_{c}}=\frac{82}{3},$ (38) $\displaystyle
d_{3}$ $\displaystyle=-1,$ $\displaystyle d_{4}$
$\displaystyle=\frac{1}{N_{c}}=\frac{1}{3}\,.$ (39)

With these rules we can find the matrix $\hat{R}$ as defined in (7).

###  3.2 Transformation matrices at one-loop

In this section we present our final result for the transformation matrices
between the BMU and JMS bases at the 1-loop level. The details of the
calculation are given in Section 4. The calculation can be split into the
three disconnected sectors VLL (Vector Left-Left), VLR (Vector Left-Right) and
SRR (Scalar Right-Right), which denote the
$\gamma_{\mu}P_{L}\otimes\gamma^{\mu}P_{L}$,
$\gamma_{\mu}P_{L}\otimes\gamma^{\mu}P_{R}$ and $P_{R}\otimes P_{R}$ Dirac
structures of the involved four-fermi operators, respectively. For the BMU
operator we use the following reference ordering

VLL
$\displaystyle:\big{\\{}Q_{1},\;Q_{2},\;\;Q_{3},\;Q_{4},\;Q_{9},\;Q_{10},\;\;Q_{11},\;\;Q_{14}\big{\\}},$
(40) VLR
$\displaystyle:\big{\\{}Q_{5},\;Q_{6},\;Q_{7},\;Q_{8},\;\;Q_{12},\;Q_{13},\;\;Q_{15},\ldots,Q_{24}\big{\\}},$
(41) SRR $\displaystyle:\big{\\{}Q_{25},\ldots,Q_{40}\big{\\}}.$ (42)

For the JMS basis we use the ordering

VLL
$\displaystyle:\big{\\{}[O_{ud}^{V1,LL}]_{11ji},\;[O_{ud}^{V8,LL}]_{11ji},\;[O_{ud}^{V1,LL}]_{22ji},\;[O_{ud}^{V8,LL}]_{22ji},$
$\displaystyle\phantom{:\big{\\{}\;}[O_{dd}^{V,LL}]_{jikk},\;[O_{dd}^{V,LL}]_{jkki},\;[O_{dd}^{V,LL}]_{jiii},\;[O_{dd}^{V,LL}]_{jijj}\big{\\}},$
(43) VLR
$\displaystyle:\big{\\{}[O_{du}^{V1,LR}]_{ji11},\;[O_{du}^{V8,LR}]_{ji11},\;[O_{du}^{V1,LR}]_{ji22},\;[O_{du}^{V8,LR}]_{ji22},$
$\displaystyle\phantom{:\big{\\{}\;}[O_{dd}^{V1,LR}]_{jikk},\;[O_{dd}^{V8,LR}]_{jikk},\;[O_{dd}^{V1,LR}]_{jiii},\;[O_{dd}^{V8,LR}]_{jiii},\;$
$\displaystyle\phantom{:\big{\\{}\;}[O_{dd}^{V1,LR}]_{jijj},\;[O_{dd}^{V8,LR}]_{jijj},\;[O_{uddu}^{V1,LR}]_{1ji1}^{\dagger},\;[O_{uddu}^{V8,LR}]_{1ji1}^{\dagger},\;$
$\displaystyle\phantom{:\big{\\{}\;}[O_{uddu}^{V1,LR}]_{2ji2}^{\dagger},\;[O_{uddu}^{V8,LR}]_{2ji2}^{\dagger},[O_{dd}^{V1,LR}]_{jkki},\;[O_{dd}^{V8,LR}]_{jkki}\big{\\}},$
(44) SRR
$\displaystyle:\big{\\{}[O_{dd}^{S1,RR}]_{jiii},\;[O_{dd}^{S8,RR}]_{jiii},\;[O_{dd}^{S1,RR}]_{jijj},\;[O_{dd}^{S8,RR}]_{jijj},\;$
$\displaystyle\phantom{:\big{\\{}\;}[O_{ud}^{S1,RR}]_{11ji},\;[O_{ud}^{S8,RR}]_{11ji},\;[O_{uddu}^{S1,RR}]_{1ij1},\;[O_{uddu}^{S8,RR}]_{1ij1},$
$\displaystyle\phantom{:\big{\\{}\;}[O_{ud}^{S1,RR}]_{22ji},\;[O_{ud}^{S8,RR}]_{22ji},\;[O_{uddu}^{S1,RR}]_{2ij2},\;[O_{uddu}^{S8,RR}]_{2ij2},$
$\displaystyle\phantom{:\big{\\{}\;}[O_{dd}^{S1,RR}]_{jikk},\;[O_{dd}^{S8,RR}]_{jikk},\;[O_{dd}^{S1,RR}]_{jkki},\;[O_{dd}^{S8,RR}]_{jkki}\big{\\}}.$
(45)

At tree-level the transformation matrix $\hat{R}^{(0)}$ reads

$\displaystyle\hat{R}^{(0)}$
$\displaystyle=\begin{pmatrix}\hat{R}^{(0)}_{\text{VLL}}&0_{8\times
16}&0_{8\times 16}\\\ 0_{16\times 8}&\hat{R}^{(0)}_{\text{VLR}}&0_{16\times
16}\\\ 0_{16\times 8}&0_{16\times
16}&\hat{R}^{(0)}_{\text{SRR}}\end{pmatrix}\,,$ (46)

$\hat{R}^{(0)}_{\text{VLL}}=\begin{pmatrix}1&0&0&0&0&0&0&0\\\\[2.84544pt]
-\frac{1}{6}&\frac{1}{2}&0&0&0&0&0&0\\\\[2.84544pt]
-1&0&\frac{1}{3}&0&\frac{2}{3}&0&0&0\\\\[2.84544pt]
\frac{1}{6}&-\frac{1}{2}&-\frac{1}{18}&\frac{1}{6}&-\frac{1}{9}&\frac{1}{3}&0&0\\\\[2.84544pt]
0&0&\frac{2}{3}&0&-\frac{2}{3}&0&-1&0\\\\[2.84544pt]
0&0&0&\frac{2}{3}&0&-\frac{2}{3}&-1&0\\\\[2.84544pt]
0&0&0&0&0&0&\frac{1}{2}&\frac{1}{2}\\\\[2.84544pt]
0&0&0&0&0&0&\frac{1}{2}&-\frac{1}{2}\end{pmatrix}\,,\qquad\hat{R}^{(0)}_{\text{SRR}}=\begin{pmatrix}\hat{A}_{\text{SRR}}^{(0)}&0_{2\times
2}&0_{2\times 4}&0_{2\times 4}&0_{2\times 4}\\\\[2.84544pt] 0_{2\times
2}&\hat{A}_{\text{SRR}}^{(0)}&0_{2\times 4}&0_{2\times 4}&0_{2\times
4}\\\\[2.84544pt] 0_{4\times 2}&0_{4\times
2}&\hat{B}_{\text{SRR}}^{(0)}&0_{4\times 4}&0_{4\times 4}\\\\[2.84544pt]
0_{4\times 2}&0_{4\times 2}&0_{4\times
4}&\hat{B}_{\text{SRR}}^{(0)}&0_{4\times 4}\\\\[2.84544pt] 0_{4\times
2}&0_{4\times 2}&0_{4\times 4}&0_{4\times
4}&\hat{B}_{\text{SRR}}^{(0)}\end{pmatrix}\,\,,$ (47)

$\hat{A}_{\text{SRR}}^{(0)}=\begin{pmatrix}1&0\\\\[2.84544pt]
-\frac{5}{12}&\frac{1}{16}\end{pmatrix}\,,\qquad\hat{B}_{\text{SRR}}^{(0)}=\begin{pmatrix}0&1&0&0\\\\[2.84544pt]
\frac{1}{2}&-\frac{1}{6}&0&0\\\\[2.84544pt]
-\frac{1}{2}&0&\frac{1}{8}&0\\\\[2.84544pt]
\frac{1}{12}&-\frac{1}{4}&-\frac{1}{48}&\frac{1}{16}\end{pmatrix}\,,$ (48)

$\hat{R}^{(0)}_{\text{VLR}}=\setcounter{MaxMatrixCols}{16}\begin{pmatrix}\frac{1}{6}&0&\frac{1}{3}&0&0&0&0&0&0&\frac{1}{2}&0&0&0&0&0&0\\\\[2.84544pt]
-\frac{1}{36}&\frac{1}{12}&-\frac{1}{18}&\frac{1}{6}&0&0&0&0&\frac{1}{4}&-\frac{1}{12}&0&0&0&0&0&0\\\\[2.84544pt]
\frac{1}{6}&0&\frac{1}{3}&0&0&0&0&0&0&-\frac{1}{2}&0&0&0&0&0&0\\\\[2.84544pt]
-\frac{1}{36}&\frac{1}{12}&-\frac{1}{18}&\frac{1}{6}&0&0&0&0&-\frac{1}{4}&\frac{1}{12}&0&0&0&0&0&0\\\\[2.84544pt]
\frac{2}{3}&0&-\frac{2}{3}&0&0&-1&0&0&0&0&0&0&0&0&0&0\\\\[2.84544pt]
-\frac{1}{9}&\frac{1}{3}&\frac{1}{9}&-\frac{1}{3}&-\frac{1}{2}&\frac{1}{6}&0&0&0&0&0&0&0&0&0&0\\\\[2.84544pt]
0&0&0&0&0&\frac{1}{2}&0&\frac{1}{2}&0&0&0&0&0&0&0&0\\\\[2.84544pt]
0&0&0&0&\frac{1}{4}&-\frac{1}{12}&\frac{1}{4}&-\frac{1}{12}&0&0&0&0&0&0&0&0\\\\[2.84544pt]
0&0&0&0&0&\frac{1}{2}&0&-\frac{1}{2}&0&0&0&0&0&0&0&0\\\\[2.84544pt]
0&0&0&0&\frac{1}{4}&-\frac{1}{12}&-\frac{1}{4}&\frac{1}{12}&0&0&0&0&0&0&0&0\\\\[2.84544pt]
0&0&0&0&0&0&0&0&0&0&-2&0&0&0&0&0\\\\[2.84544pt]
0&0&0&0&0&0&0&0&0&0&\frac{1}{3}&-1&0&0&0&0\\\\[2.84544pt]
0&0&0&0&0&0&0&0&0&0&0&0&-2&0&0&0\\\\[2.84544pt]
0&0&0&0&0&0&0&0&0&0&0&0&\frac{1}{3}&-1&0&0\\\\[2.84544pt]
0&0&0&0&0&0&0&0&0&0&0&0&0&0&-2&0\\\\[2.84544pt]
0&0&0&0&0&0&0&0&0&0&0&0&0&0&\frac{1}{3}&-1\end{pmatrix}\,.$ (49)

At 1-loop level the corrections due to the EV operators are given by the
matrix $\hat{R}^{(1)}$

$\displaystyle\hat{R}^{(1)}$
$\displaystyle=\begin{pmatrix}\hat{R}^{(1)}_{\text{VLL}}&R^{(1)}_{8\times
16}&0_{8\times 16}\\\ 0_{16\times 8}&\hat{R}^{(1)}_{\text{VLR}}&0_{16\times
16}\\\ 0_{16\times 8}&0_{16\times
16}&\hat{R}^{(1)}_{\text{SRR}}\end{pmatrix}\,,$ (50)

$\hat{R}^{(1)}_{\text{VLL}}=\begin{pmatrix}0&0&0&0&0&0&0&0\\\\[2.84544pt]
0&0&\frac{1}{18}&-\frac{1}{6}&0&0&0&0\\\\[2.84544pt]
0&0&0&0&0&0&0&0\\\\[2.84544pt]
0&0&-\frac{1}{18}&\frac{1}{6}&0&0&0&0\\\\[2.84544pt]
0&0&0&0&0&0&0&0\\\\[2.84544pt] 0&0&-\frac{1}{3}&1&0&0&0&0\\\\[2.84544pt]
0&0&0&0&0&0&0&0\\\\[2.84544pt]
0&0&0&0&0&0&0&0\end{pmatrix}\,,\qquad\hat{R}^{(1)}_{8\times
16}=\begin{pmatrix}0&0&\dots&0\\\\[2.84544pt]
\frac{1}{18}&-\frac{1}{6}&\dots&0\\\\[2.84544pt] 0&0&\dots&0\\\\[2.84544pt]
-\frac{1}{18}&\frac{1}{6}&\dots&0\\\\[2.84544pt] 0&0&\dots&0\\\\[2.84544pt]
-\frac{1}{3}&1&\dots&0\\\\[2.84544pt] 0&0&\dots&0\\\\[2.84544pt]
0&0&\dots&0\end{pmatrix}\,,\qquad\hat{R}^{(1)}_{\text{VLR}}=0_{16\times
16}\,,$ (51)

$\hat{R}^{(1)}_{\text{SRR}}=\begin{pmatrix}\hat{A}^{(1)}_{\text{SRR}}&0_{2\times
2}&0_{2\times 4}&0_{2\times 4}&0_{2\times 4}\\\\[2.84544pt] 0_{2\times
2}&\hat{A}^{(1)}_{\text{SRR}}&0_{2\times 4}&0_{2\times 4}&0_{2\times
4}\\\\[2.84544pt] 0_{4\times 2}&0_{4\times
2}&\hat{B}^{(1)}_{\text{SRR}}&0_{4\times 4}&0_{4\times 4}\\\\[2.84544pt]
0_{4\times 2}&0_{4\times 2}&0_{4\times
4}&\hat{B}^{(1)}_{\text{SRR}}&0_{4\times 4}\\\\[2.84544pt] 0_{4\times
2}&0_{4\times 2}&0_{4\times 4}&0_{4\times
4}&\hat{B}^{(1)}_{\text{SRR}}\end{pmatrix}\,,\quad\begin{array}[]{c}\hat{A}^{(1)}_{\text{SRR}}=\begin{pmatrix}0&0\\\\[2.84544pt]
-\frac{37}{24}&-\frac{29}{96}\end{pmatrix}\,,\\\
\quad\hat{B}^{(1)}_{\text{SRR}}=\begin{pmatrix}0&0&0&0\\\\[2.84544pt]
0&0&0&0\\\\[2.84544pt]
-\frac{7}{12}&-\frac{17}{4}&-\frac{5}{48}&-\frac{3}{16}\\\\[2.84544pt]
-\frac{55}{36}&-\frac{13}{12}&-\frac{11}{144}&-\frac{1}{48}\end{pmatrix}\,.\end{array}$
(52)

Here, we have used $N_{f}=5,N_{d}=3$ and $N_{u}=2$.

##  4 Evanescent operators

In this section, we define and calculate the EV operators which in turn gives
us the matrix $\hat{R}^{(1)}$.

###  4.1 Definition of EV operators in the VLL sector

At the 1-loop level, we need to add an EV contribution to a tree-level basis
transformation rule if it involves a Fierz transformation. On the other hand,
we do not need an EV contribution for the cases in which no Fierz is required
for the tree-level basis change. In the VLL sector, there are total eight
independent operators where only five of them involve Fierz relation (105) for
the change of basis. The corresponding EV operators $E_{I}^{\text{VLL}},I=1-5$
are defined by the relations:

$\displaystyle[O_{ud}^{V1,LL}]_{11ji}$
$\displaystyle\stackrel{{\scriptstyle\mathcal{F}}}{{=}}$ $\displaystyle
Q_{1}+E_{1}^{\text{VLL}}\,,$ (53) $\displaystyle[O_{ud}^{V8,LL}]_{11ji}$
$\displaystyle\stackrel{{\scriptstyle\mathcal{F}}}{{=}}$
$\displaystyle-\frac{1}{6}Q_{1}+\frac{1}{2}Q_{2}+E_{2}^{\text{VLL}}\,,$ (54)
$\displaystyle[O_{ud}^{V1,LL}]_{22ji}$
$\displaystyle\stackrel{{\scriptstyle\mathcal{F}}}{{=}}$ $\displaystyle-
Q_{1}+\frac{1}{3}Q_{3}+\frac{2}{3}Q_{9}+E_{3}^{\text{VLL}}\,,$ (55)
$\displaystyle[O_{ud}^{V8,LL}]_{22ji}$
$\displaystyle\stackrel{{\scriptstyle\mathcal{F}}}{{=}}$
$\displaystyle\frac{1}{6}Q_{1}-\frac{1}{2}Q_{2}-\frac{1}{18}Q_{3}+\frac{1}{6}Q_{4}-\frac{1}{9}Q_{9}+\frac{1}{3}Q_{10}+E_{4}^{\text{VLL}}\,,$
(56) $\displaystyle[O_{dd}^{V,LL}]_{jikk}$ $\displaystyle=$
$\displaystyle\frac{2}{3}Q_{3}-\frac{2}{3}Q_{9}-Q_{11}\,,$ (57)
$\displaystyle[O_{dd}^{V,LL}]_{jkki}$
$\displaystyle\stackrel{{\scriptstyle\mathcal{F}}}{{=}}$
$\displaystyle\frac{2}{3}Q_{4}-\frac{2}{3}Q_{10}-Q_{11}+E_{5}^{\text{VLL}}\,,$
(58) $\displaystyle[O_{dd}^{V,LL}]_{jiii}$ $\displaystyle=$
$\displaystyle\frac{1}{2}Q_{11}+\frac{1}{2}Q_{14}\,,$ (59)
$\displaystyle[O_{dd}^{V,LL}]_{jijj}$ $\displaystyle=$
$\displaystyle\frac{1}{2}Q_{11}-\frac{1}{2}Q_{14}.$ (60)

Here $\mathcal{F}$ indicates that the Fierz identity (105) is needed for the
change of basis.

###  4.2 Calculation of the EV operators in the VLL sectors

Now we are in position to use the rules presented in Sec. 3.1 to obtain
$E_{1}^{\text{VLL}}$ -$E_{5}^{\text{VLL}}$ which contribute to
$\hat{R}^{(1)}$. In order to use the rules of Sec.3.1 first we need to express
the JMS operators on the l.h.s. in terms of the $\widetilde{Q}_{I}$ operators.
In general, there are three categories of operators as discussed in Sec. 2.3.

####  4.2.1 Operators requiring no Fierz

Since the following set of operators in the VLL sector do not require Fierz
transformations for the basis change

$[O_{dd}^{V,LL}]_{jikk}\,,[O_{dd}^{V,LL}]_{jiii}\,,[O_{dd}^{V,LL}]_{jijj}\,,$
(61)

there are no EV operator contributions at the 1-loop level basis
transformation given by (57),(59) and (60). Hence, the corresponding entries
in the matrix $\hat{R}^{(1)}$ vanish.

####  4.2.2 Operators requiring Fierz but no EV shifts

There are two operators in the JMS basis which require Fierz transformation
but the EV contribution still vanish. The tree-level transformations for these
operators are given by (53) and (55). To see this, one has to express the JMS
operators on the LHS in terms of the $\widetilde{Q}_{I}$ operators, doing so
we obtain

$\displaystyle E_{1}^{\text{VLL}}$ $\displaystyle=$
$\displaystyle[O_{ud}^{V1,LL}]_{11ji}-Q_{1}~{}=~{}\widetilde{Q}_{1}-Q_{1}=0\,,$
(62) $\displaystyle E_{3}^{\text{VLL}}$ $\displaystyle=$
$\displaystyle[O_{ud}^{V1,LL}]_{22ji}-\left(-Q_{1}+\frac{1}{3}Q_{3}+\frac{2}{3}Q_{9}\right)~{}=~{}\widetilde{Q}_{1}-Q_{1}=~{}0.$
(63)

Here the shift in $\widetilde{Q}_{1}-Q_{1}$ is given by the rule (11).

####  4.2.3 Operators requiring Fierz and EV shifts

Finally, we turn to the cases for which Fierz transformation at the tree-level
as well as the EV contributions at the 1-loop level are necessary for the
basis transformation. The tree-level transformations can be read from
Eqs.(54), (56) and (58). The EV operators are then given by

$\displaystyle E_{2}^{\text{VLL}}$ $\displaystyle=$
$\displaystyle[O_{ud}^{V8,LL}]_{11ji}-\left(-\frac{1}{6}Q_{1}+\frac{1}{2}Q_{2}\right)$
(64) $\displaystyle=$
$\displaystyle\frac{1}{6}(Q_{1}-\widetilde{Q}_{1})-\frac{1}{2}(Q_{2}-\widetilde{Q}_{2})~{}=~{}-\frac{1}{6}\frac{\alpha_{s}}{4\pi}P\,,$
$\displaystyle E_{4}^{\text{VLL}}$ $\displaystyle=$
$\displaystyle=[O_{ud}^{V8,LL}]_{22ji}-\left(\frac{1}{6}Q_{1}-\frac{1}{2}Q_{2}-\frac{1}{18}Q_{3}+\frac{1}{6}Q_{4}-\frac{1}{9}Q_{9}+\frac{1}{3}Q_{10}\right)$
(65) $\displaystyle=$
$\displaystyle\frac{1}{6}(\widetilde{Q}_{1}-Q_{1})+\frac{1}{2}(Q_{2}-\widetilde{Q}_{2})=\frac{1}{6}\frac{\alpha_{s}}{4\pi}P\,,$
$\displaystyle E_{5}^{\text{VLL}}$ $\displaystyle=$
$\displaystyle[O_{dd}^{V,LL}]_{jkki}-\left(\frac{2}{3}Q_{4}-\frac{2}{3}Q_{10}-Q_{11}\right)$
(66) $\displaystyle=$
$\displaystyle{\frac{2}{3}(\widetilde{Q}_{4}-Q_{4})-\frac{2}{3}(\widetilde{Q}_{10}-Q_{10})}$
$\displaystyle=$
$\displaystyle{\frac{2N_{f}+N_{d}-2N_{u}}{9}\frac{\alpha_{s}}{4\pi}P}.$

Here the 1-loop shifts are given by the rules in (11),(13),(14) and (15),
respectively.

###  4.3 Definition of EV operators in the SRR sector

In this case, in addition to the color identity (109), we need the Fierz
relations given in (103) and (107) 777Note that here we have used the
definition $\sigma_{\mu\nu}=\frac{i}{2}[\gamma_{\mu},\gamma_{\nu}]$. Also we
define the operators $Q_{2}^{\text{SRR},i}$ and $Q_{3}^{\text{SRR},Q}$,
$Q_{4}^{\text{SRR},Q}$ with an additional negative sign as compared to
Ref.[11].  The 1-loop basis transformations including the EV operators
$E_{I}^{\text{SRR}}$ read

$\displaystyle[O_{dd}^{S1,RR}]_{jiii}$ $\displaystyle=$
$\displaystyle{Q_{2}^{\text{SRR},i}}\,,$ (67)
$\displaystyle[O_{dd}^{S8,RR}]_{jiii}$
$\displaystyle\stackrel{{\scriptstyle\mathcal{F}}}{{=}}$
$\displaystyle-\frac{5}{12}{Q_{2}^{\text{SRR},i}}+\frac{1}{16}{Q_{4}^{\text{SRR},i}}+E^{\text{SRR,i}}.$
(68)

Similar relations hold for $jiii\to jijj$. We note that a separate evanescent
operator $E^{\text{SRR,j}}$ is needed for this relation.

For the SRR,Q operators we find:

$\displaystyle[O_{ud}^{S1,RR}]_{11ji}$ $\displaystyle=$ $\displaystyle
Q_{2}^{\text{SRR},u}\,,$ (69) $\displaystyle[O_{ud}^{S8,RR}]_{11ji}$
$\displaystyle=$
$\displaystyle\frac{1}{2}Q_{1}^{\text{SRR},u}-\frac{1}{6}Q_{2}^{\text{SRR},u}\,,$
(70) $\displaystyle[O_{uddu}^{S1,RR}]_{1ij1}$
$\displaystyle\stackrel{{\scriptstyle\mathcal{F}}}{{=}}$
$\displaystyle-\frac{1}{2}Q_{1}^{\text{SRR},u}+\frac{1}{8}Q_{3}^{\text{SRR},u}+E_{1}^{\text{SRR,u}}\,,$
(71) $\displaystyle[O_{uddu}^{S8,RR}]_{1ij1}$
$\displaystyle\stackrel{{\scriptstyle\mathcal{F}}}{{=}}$
$\displaystyle\frac{1}{12}Q_{1}^{\text{SRR},u}-\frac{1}{4}Q_{2}^{\text{SRR},u}-\frac{1}{48}Q_{3}^{\text{SRR},u}+\frac{1}{16}Q_{4}^{\text{SRR},u}+E_{2}^{\text{SRR,u}}.$
(72)

Similar relations hold for $1\to 2$ and $u\to c$ on the l.h.s. and r.h.s.,
respectively. Finally,

$\displaystyle[O_{dd}^{S1,RR}]_{jikk}$ $\displaystyle=$ $\displaystyle
Q_{2}^{\text{SRR},d_{k}}\,,$ (73) $\displaystyle[O_{dd}^{S8,RR}]_{jikk}$
$\displaystyle=$
$\displaystyle\frac{1}{2}Q_{1}^{\text{SRR},d_{k}}-\frac{1}{6}Q_{2}^{\text{SRR},d_{k}}\,,$
(74) $\displaystyle[O_{dd}^{S1,RR}]_{jkki}$
$\displaystyle\stackrel{{\scriptstyle\mathcal{F}}}{{=}}$
$\displaystyle-\frac{1}{2}Q_{1}^{\text{SRR},d_{k}}+\frac{1}{8}Q_{3}^{\text{SRR},d_{k}}+E_{1}^{\text{SRR,$d_{k}$}}\,,$
(75) $\displaystyle[O_{dd}^{S8,RR}]_{jkki}$
$\displaystyle\stackrel{{\scriptstyle\mathcal{F}}}{{=}}$
$\displaystyle\frac{1}{12}Q_{1}^{\text{SRR},d_{k}}-\frac{1}{4}Q_{2}^{\text{SRR},d_{k}}-\frac{1}{48}Q_{3}^{\text{SRR},d_{k}}+\frac{1}{16}Q_{4}^{\text{SRR},d_{k}}+E_{2}^{\text{SRR,$d_{k}$}}.$
(76)

###  4.4 Calculation of EV operators in the SRR sector

####  4.4.1 Operators requiring no Fierz

In the SRR sector, the following operators do not require Fierz
transformations for the basis change at the tree-level

$\displaystyle[O_{dd}^{S1,RR}]_{jiii}\,,$
$\displaystyle{[O_{dd}^{S1,RR}]_{jijj}\,,}[O_{ud}^{S1,RR}]_{11ji}\,,[O_{ud}^{S8,RR}]_{11ji}\,,{[O_{ud}^{S1,RR}]_{22ji}\,,}$
(77)
$\displaystyle{[O_{ud}^{S8,RR}]_{22ji}\,,}[O_{dd}^{S1,RR}]_{jikk}\,,[O_{ud}^{S8,RR}]_{jikk}.$

Therefore, for the basis change at the 1-loop level no EV contributions are
required.

####  4.4.2 Operators requiring Fierz but no EV shifts

In this sector there are no such operators which require Fierz without having
non vanishing EV shifts at the 1-loop level.

####  4.4.3 Operators requiring Fierz and EV shifts

The SRR operators which require the Fierz relation and non vanishing EV
operators for the basis transformation are given by (68), (71), (72), (75),
and (76). The EV operators are then given by

$\displaystyle E^{\text{SRR,i}}$ $\displaystyle=$
$\displaystyle[O_{dd}^{S8,RR}]_{jiii}-\left(-\frac{5}{12}Q_{2}^{\text{SRR},i}+{\frac{1}{16}}Q_{4}^{\text{SRR},i}\right)={-}\frac{1}{2}(\widetilde{Q}_{1}^{\text{SRR},i}-Q_{1}^{\text{SRR},i})$
(78) $\displaystyle=$
$\displaystyle{-}\frac{\alpha_{s}}{4\pi}\left(\frac{37}{24}Q_{2}^{\text{SRR},i}+\frac{29}{96}Q_{4}^{\text{SRR},i}\right)\,,$
$\displaystyle E_{1}^{\text{SRR,u}}$ $\displaystyle=$
$\displaystyle[O_{uddu}^{S1,RR}]_{1ij1}-\left(-\frac{1}{2}Q_{1}^{\text{SRR},u}+\frac{1}{8}Q_{3}^{\text{SRR},u}\right)$
(79) $\displaystyle=$
$\displaystyle-\frac{1}{2}(\widetilde{Q}_{1}^{\text{SRR},u}-Q_{1}^{\text{SRR},u})+\frac{1}{8}(\widetilde{Q}_{3}^{\text{SRR},u}-Q_{3}^{\text{SRR},u})$
$\displaystyle=$
$\displaystyle\frac{\alpha_{s}}{4\pi}\sum_{k=1,2,3,4}p_{k}Q_{k}^{\text{SRR,u}}\,,$
$\displaystyle E_{2}^{\text{SRR,u}}$ $\displaystyle=$
$\displaystyle[O_{uddu}^{S8,RR}]_{1ij1}-\left(\frac{1}{12}Q_{1}^{\text{SRR},u}-\frac{1}{4}Q_{2}^{\text{SRR},u}-\frac{1}{48}Q_{3}^{\text{SRR},u}+\frac{1}{16}Q_{4}^{\text{SRR},u}\right)$
(80) $\displaystyle=$
$\displaystyle\frac{1}{12}(\widetilde{Q}_{1}^{\text{SRR},u}-Q_{1}^{\text{SRR},u})-\frac{1}{4}(\widetilde{Q}_{2}^{\text{SRR},u}-Q_{2}^{\text{SRR},u})$
$\displaystyle-$
$\displaystyle\frac{1}{48}(\widetilde{Q}_{3}^{\text{SRR},u}-Q_{3}^{\text{SRR},u})+\frac{1}{16}(\widetilde{Q}_{4}^{\text{SRR},u}-Q_{4}^{\text{SRR},u})$
$\displaystyle=$
$\displaystyle\frac{\alpha_{s}}{4\pi}\sum_{k=1,2,3,4}q_{k}Q_{k}^{\text{SRR,u}}.$

The coefficients $p_{k}$ and $q_{k}$ are found to be

$\displaystyle p_{1}$ $\displaystyle=$ $\displaystyle-\frac{7}{12}\,,\,\
p_{2}=-\frac{17}{4}\,,\,\ p_{3}=-\frac{5}{48}\,,\,\ p_{4}=-\frac{3}{16}\,,$
(81) $\displaystyle q_{1}$ $\displaystyle=$ $\displaystyle-\frac{55}{36}\,,\,\
q_{2}=-\frac{13}{12}\,,\,\ q_{3}=-\frac{11}{144}\,,\,\ q_{4}=-\frac{1}{48}\,.$
(82)

The evanescent operators $E^{\text{SRR,j}}$ and
$E_{1,2}^{\text{SRR,c}}\,,E_{1,2}^{\text{SRR,$d_{k}$}}$ follow from the
corresponding $E^{\text{SRR,i}}$ and $E_{1,2}^{\text{SRR,u}}$ by the
corresponding flavour replacements.

In the above calculation, the shifts
$Q_{1}^{\text{SRR},D}-\widetilde{Q}_{1}^{\text{SRR},D}$, and
$Q_{I}^{\text{SRR},Q}-\widetilde{Q}_{I}^{\text{SRR},Q}$ for $Q=u$ or $d_{k}$
are obtained using the rules (20) and (28) - (31), respectively. It is worth
noting at the 1-loop QCD only current-current insertions are involved for the
SRR operators in obtaining these rules. Therefore flavour structure of the
operators is immaterial. For instance, the operators
$[O_{uddu}^{S1,RR}]_{1ij1}$ and $Q_{2}^{\text{SRR,u}}$ having same color and
Lorentz structures can be treated on the same footing even though they have
different flavour structures.

###  4.5 Definition of EV operators in the VLR sector

In this subsection we turn our attention to the VLR sector and define the
corresponding evanescent operators. Using the Fierz relation in eq. (106) as
well as colour relations one finds:

$\displaystyle[O_{du}^{V1,LR}]_{ji11}$
$\displaystyle\stackrel{{\scriptstyle\mathcal{F}}}{{=}}$
$\displaystyle\frac{1}{6}Q_{5}+\frac{1}{3}Q_{7}+\frac{1}{2}Q_{18}+E_{1}^{\text{VLR}}\,,$
$\displaystyle[O_{du}^{V8,LR}]_{ji11}$
$\displaystyle\stackrel{{\scriptstyle\mathcal{F}}}{{=}}$
$\displaystyle-\frac{1}{36}Q_{5}+\frac{1}{12}Q_{6}-\frac{1}{18}Q_{7}+\frac{1}{6}Q_{8}+\frac{1}{4}Q_{17}-\frac{1}{12}Q_{18}+E_{2}^{\text{VLR}}\,,$
$\displaystyle[O_{du}^{V1,LR}]_{ji22}$
$\displaystyle\stackrel{{\scriptstyle\mathcal{F}}}{{=}}$
$\displaystyle\frac{1}{6}Q_{5}+\frac{1}{3}Q_{7}-\frac{1}{2}Q_{18}+E_{3}^{\text{VLR}}\,,$
$\displaystyle[O_{du}^{V8,LR}]_{ji22}$
$\displaystyle\stackrel{{\scriptstyle\mathcal{F}}}{{=}}$
$\displaystyle-\frac{1}{36}Q_{5}+\frac{1}{12}Q_{6}-\frac{1}{18}Q_{7}+\frac{1}{6}Q_{8}-\frac{1}{4}Q_{17}+\frac{1}{12}Q_{18}+E_{4}^{\text{VLR}}\,,$
$\displaystyle[O_{dd}^{V1,LR}]_{jikk}$ $\displaystyle=$
$\displaystyle\frac{2}{3}Q_{5}-\frac{2}{3}Q_{7}-Q_{13}\,,$
$\displaystyle[O_{dd}^{V8,LR}]_{jikk}$
$\displaystyle\stackrel{{\scriptstyle\mathcal{F}}}{{=}}$
$\displaystyle-\frac{1}{9}Q_{5}+\frac{1}{3}Q_{6}+\frac{1}{9}Q_{7}-\frac{1}{3}Q_{8}-\frac{1}{2}Q_{12}+\frac{1}{6}Q_{13}+E_{5}^{\text{VLR}}\,,$
$\displaystyle[O_{dd}^{V1,LR}]_{jiii}$ $\displaystyle=$
$\displaystyle\frac{1}{2}Q_{13}+\frac{1}{2}Q_{16}\,,$
$\displaystyle[O_{dd}^{V8,LR}]_{jiii}$ $\displaystyle=$
$\displaystyle\frac{1}{4}Q_{12}-\frac{1}{12}Q_{13}+\frac{1}{4}Q_{15}-\frac{1}{12}Q_{16}\,,$
$\displaystyle[O_{dd}^{V1,LR}]_{jijj}$ $\displaystyle=$
$\displaystyle\frac{1}{2}Q_{13}-\frac{1}{2}Q_{16}\,,$
$\displaystyle[O_{dd}^{V8,LR}]_{jijj}$ $\displaystyle=$
$\displaystyle\frac{1}{4}Q_{12}-\frac{1}{12}Q_{13}-\frac{1}{4}Q_{15}+\frac{1}{12}Q_{16}\,,$
$\displaystyle[O_{uddu}^{V1,LR}]_{1ji1}^{\dagger}$ $\displaystyle=$
$\displaystyle-2Q_{19}\,,$ $\displaystyle[O_{uddu}^{V8,LR}]_{1ji1}^{\dagger}$
$\displaystyle=$ $\displaystyle\frac{1}{3}Q_{19}-Q_{20}\,,$
$\displaystyle[O_{uddu}^{V1,LR}]_{2ji2}^{\dagger}$ $\displaystyle=$
$\displaystyle-2Q_{21}\,,$ $\displaystyle[O_{uddu}^{V8,LR}]_{2ji2}^{\dagger}$
$\displaystyle=$ $\displaystyle\frac{1}{3}Q_{21}-Q_{22}\,,$
$\displaystyle[O_{dd}^{V1,LR}]_{jkki}$ $\displaystyle=$
$\displaystyle-2Q_{23}\,,$ $\displaystyle[O_{dd}^{V8,LR}]_{jkki}$
$\displaystyle=$ $\displaystyle\frac{1}{3}Q_{23}-Q_{24}.$ (83)

In the VLR sector, the following set operators do not require Fierz
transformations for the basis change at the tree-level

$\displaystyle[O_{dd}^{V1,LR}]_{jikk}$
$\displaystyle\,,[O_{dd}^{V1,LR}]_{jiii}\,,[O_{dd}^{V8,LR}]_{jiii}\,,[O_{dd}^{V1,LR}]_{jijj}\,,[O_{dd}^{V8,LR}]_{jijj}\,,[O_{uddu}^{V1,LR}]_{1ji1}^{\dagger}\,,$
$\displaystyle[O_{uddu}^{V8,LR}]_{1ji1}^{\dagger}$
$\displaystyle\,,[O_{uddu}^{V1,LR}]_{2ji2}^{\dagger}\,,[O_{uddu}^{V8,LR}]_{2ji2}^{\dagger}\,,[O_{dd}^{V1,LR}]_{jkki}\,,[O_{dd}^{V8,LR}]_{jkki}\,.$
(84)

Therefore, for the basis change at the 1-loop level no EV contributions are
required. The rest of the operators in the VLR sector requiring Fierz are

$[O_{du}^{V1,LR}]_{ji11}\,,[O_{du}^{V8,LR}]_{ji11}\,,[O_{du}^{V1,LR}]_{ji22}\,,[O_{du}^{V8,LR}]_{ji22}\,,[O_{dd}^{V8,LR}]_{jikk}\,.$
(85)

However, as discussed in Subsec. 3.1 the corresponding EV vanish:

$E_{1}^{\text{VLR}}=E_{2}^{\text{VLR}}=E_{3}^{\text{VLR}}=E_{4}^{\text{VLR}}=E_{5}^{\text{VLR}}=0.$
(86)

##  5 Conclusions

We have presented a simple recipe to perform one-loop basis transformations
involving evanescent operators. The procedure consists of computing the
commutator of a one-loop $(L)$ correction using dimensional regularisation and
a Fierz ($\mathcal{F}$) transformation of a given operator $Q$, which in all
generality is non-vanishing:

$[L,\mathcal{F}]Q\neq 0\,.$ (87)

The presented method has been already used successfully in several contexts
such as NLO basis transformations [15, 11, 24], one-loop matching calculations
[25] as well as in several two-loop calculations [10]. But it has not been
presented in any detail and in particular in this generality in the literature
so far. The present paper should help to clarify possible issues involving
evanescent operators. In the coming years one-loop matching and two-loop
running effects will become more important in NP analyses than they are now.

We illustrated the outlined procedure by computing explicitly the complete
one-loop basis change from the BMU to the JMS basis at
$\mathcal{O}(\alpha_{s})$ and this example should allow the reader to perform
the transformation between different bases. In this context our method will
serve as a simple tool to perform one-loop basis transformations. One
particular example would be the basis change to the CMM (Chetyrkin, Misiak and
Munz) basis [18], which is most suited for multi-loop computations.

Since the one-loop basis change consists of a series of simple algebraic
manipulations, it would be interesting to automate this procedure. After
having computed all one-loop corrections to the operators in question a simple
algorithm might be included in codes such as for example abc-eft [26].

## Acknowledgments

J. A. acknowledges financial support from the European Research Council (ERC)
under the European Union’s Horizon 2020 research and innovation programme
under grant agreement 833280 (FLAY), and from the Swiss National Science
Foundation (SNF) under contract 200020-204428. A.J.B acknowledges financial
support from the Excellence Cluster ORIGINS, funded by the Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation) under Germany´s
Excellence Strategy – EXC-2094 – 390783311. J.K. is financially supported by
the Alexander von Humboldt Foundation’s postdoctoral research fellowship.

##  Appendix A $\Delta F=1$ BMU basis for $N_{f}=5$

In this Appendix we collect the full set of BMU operators.

We start the list with the vector operators, where the first ten operators are
the well known SM operators $Q_{1}-Q_{10}$:

$\displaystyle Q_{1}$
$\displaystyle=Q_{1}^{\text{VLL},u}=(\bar{d}_{j}^{\alpha}\gamma_{\mu}P_{L}u^{\beta})(\bar{u}^{\beta}\gamma^{\mu}P_{L}d_{i}^{\alpha}),$
(88) $\displaystyle Q_{2}$
$\displaystyle=Q_{2}^{\text{VLL},u}=(\bar{d}_{j}^{\alpha}\gamma_{\mu}P_{L}u^{\alpha})(\bar{u}^{\beta}\gamma^{\mu}P_{L}d_{i}^{\beta}),$
$\displaystyle Q_{3}$
$\displaystyle=(\bar{d}_{j}^{\alpha}\gamma_{\mu}P_{L}d_{i}^{\alpha})\\!\sum_{q}(\bar{q}^{\beta}\gamma^{\mu}P_{L}\,q^{\beta}),\qquad$
$\displaystyle Q_{4}$
$\displaystyle=(\bar{d}_{j}^{\alpha}\gamma_{\mu}P_{L}d_{i}^{\beta})\\!\sum_{q}(\bar{q}^{\beta}\gamma^{\mu}P_{L}\,q^{\alpha}),$
(89) $\displaystyle Q_{5}$
$\displaystyle=(\bar{d}_{j}^{\alpha}\gamma_{\mu}P_{L}d_{i}^{\alpha})\\!\sum_{q}(\bar{q}^{\beta}\gamma^{\mu}P_{R}\,q^{\beta}),$
$\displaystyle Q_{6}$
$\displaystyle=(\bar{d}_{j}^{\alpha}\gamma_{\mu}P_{L}d_{i}^{\beta})\\!\sum_{q}(\bar{q}^{\beta}\gamma^{\mu}P_{R}\,q^{\alpha}),$
$\displaystyle Q_{7}$
$\displaystyle=\frac{3}{2}\,(\bar{d}_{j}^{\alpha}\gamma_{\mu}P_{L}d_{i}^{\alpha})\\!\sum_{q}\\!Q_{q}\,(\bar{q}^{\beta}\gamma^{\mu}P_{R}\,q^{\beta}),\qquad$
$\displaystyle Q_{8}$
$\displaystyle=\frac{3}{2}\,(\bar{d}_{j}^{\alpha}\gamma_{\mu}P_{L}d_{i}^{\beta})\\!\sum_{q}\\!Q_{q}\,(\bar{q}^{\beta}\gamma^{\mu}P_{R}\,q^{\alpha}),$
(90) $\displaystyle Q_{9}$
$\displaystyle=\frac{3}{2}\,(\bar{d}_{j}^{\alpha}\gamma_{\mu}P_{L}d_{i}^{\alpha})\\!\sum_{q}\\!Q_{q}\,(\bar{q}^{\beta}\gamma^{\mu}P_{L}\,q^{\beta}),$
$\displaystyle Q_{10}$
$\displaystyle=\frac{3}{2}\,(\bar{d}_{j}^{\alpha}\gamma_{\mu}P_{L}d_{i}^{\beta})\\!\sum_{q}\\!Q_{q}\,(\bar{q}^{\beta}\gamma^{\mu}P_{L}\,q^{\alpha}).$

The NP (New Physics) vector operators in the BMU basis are given by:

$\displaystyle Q_{11}=\;Q_{1}^{\text{VLL},i+j}$
$\displaystyle=(\bar{d}_{j}^{\alpha}\gamma_{\mu}P_{L}d_{i}^{\alpha})\,\big{[}(\bar{d}_{i}^{\beta}\gamma^{\mu}P_{L}d_{i}^{\beta})+(\bar{d}_{j}^{\beta}\gamma^{\mu}P_{L}d_{j}^{\beta})\big{]},$
(91) $\displaystyle Q_{12}=\;Q_{1}^{\text{VLR},i+j}$
$\displaystyle=(\bar{d}_{j}^{\alpha}\gamma_{\mu}P_{L}d_{i}^{\beta})\,\big{[}(\bar{d}_{i}^{\beta}\gamma^{\mu}P_{R}d_{i}^{\alpha})+(\bar{d}_{j}^{\beta}\gamma^{\mu}P_{R}d_{j}^{\alpha})\big{]},$
$\displaystyle Q_{13}=\;Q_{2}^{\text{VLR},i+j}$
$\displaystyle=(\bar{d}_{j}^{\alpha}\gamma_{\mu}P_{L}d_{i}^{\alpha})\,\big{[}(\bar{d}_{i}^{\beta}\gamma^{\mu}P_{R}d_{i}^{\beta})+(\bar{d}_{j}^{\beta}\gamma^{\mu}P_{R}d_{j}^{\beta})\big{]},$
$\displaystyle Q_{14}=\;Q_{1}^{\text{VLL},i-j}$
$\displaystyle=(\bar{d}_{j}^{\alpha}\gamma_{\mu}P_{L}d_{i}^{\alpha})\,\big{[}(\bar{d}_{i}^{\beta}\gamma^{\mu}P_{L}d_{i}^{\beta})-(\bar{d}_{j}^{\beta}\gamma^{\mu}P_{L}d_{j}^{\beta})\big{]},$
(92) $\displaystyle Q_{15}=\;Q_{1}^{\text{VLR},i-j}$
$\displaystyle=(\bar{d}_{j}^{\alpha}\gamma_{\mu}P_{L}d_{i}^{\beta})\,\big{[}(\bar{d}_{i}^{\beta}\gamma^{\mu}P_{R}d_{i}^{\alpha})-(\bar{d}_{j}^{\beta}\gamma^{\mu}P_{R}d_{j}^{\alpha})\big{]},$
$\displaystyle Q_{16}=\;Q_{2}^{\text{VLR},i-j}$
$\displaystyle=(\bar{d}_{j}^{\alpha}\gamma_{\mu}P_{L}d_{i}^{\alpha})\,\big{[}(\bar{d}_{i}^{\beta}\gamma^{\mu}P_{R}d_{i}^{\beta})-(\bar{d}_{j}^{\beta}\gamma^{\mu}P_{R}d_{j}^{\beta})\big{]},$
$\displaystyle Q_{17}=\;Q_{1}^{\text{VLR},u-c}$
$\displaystyle=(\bar{d}_{j}^{\alpha}\gamma_{\mu}P_{L}d_{i}^{\beta})\,\big{[}(\bar{u}^{\beta}\gamma^{\mu}P_{R}\,u^{\alpha})-(\bar{c}^{\beta}\gamma^{\mu}P_{R}\,c^{\alpha})\big{]},$
(93) $\displaystyle Q_{18}=\;Q_{2}^{\text{VLR},u-c}$
$\displaystyle=(\bar{d}_{j}^{\alpha}\gamma_{\mu}P_{L}d_{i}^{\alpha})\,\big{[}(\bar{u}^{\beta}\gamma^{\mu}P_{R}\,u^{\beta})-(\bar{c}^{\beta}\gamma^{\mu}P_{R}\,c^{\beta})\big{]}.$

Finally, we introduce the scalar sector of the BMU basis. In the SRL sector we
use the structures

$\displaystyle Q_{1}^{\text{SRL},Q}$
$\displaystyle=(\bar{d}_{j}^{\alpha}P_{R}d_{i}^{\beta})\,(\bar{Q}^{\beta}P_{L}\,Q^{\alpha}),$
$\displaystyle Q_{2}^{\text{SRL},Q}$
$\displaystyle=(\bar{d}_{j}^{\alpha}P_{R}d_{i}^{\alpha})\,(\bar{Q}^{\beta}P_{L}\,Q^{\beta}),$
(94)

which define the operators

$\displaystyle(Q_{19},\,Q_{20})$
$\displaystyle=(Q_{1}^{\text{SRL},u},\,Q_{2}^{\text{SRL},u}),\qquad\qquad$
(95) $\displaystyle(Q_{21},\,Q_{22})$
$\displaystyle=(Q_{1}^{\text{SRL},c},\,Q_{2}^{\text{SRL},c}),\qquad\qquad$
$\displaystyle(Q_{23},\,Q_{24})$
$\displaystyle=(Q_{1}^{\text{SRL},d_{k}},\,Q_{2}^{\text{SRL},d_{k}}).$

In the SRR sector with three equal quarks we introduce for completeness the
redundant structures

$\displaystyle Q_{1}^{\text{SRR},i}$
$\displaystyle=(\bar{d}_{j}^{\alpha}P_{R}\,d_{i}^{\beta})\,(\bar{d}_{i}^{\beta}P_{R}\,d_{i}^{\alpha}),$
$\displaystyle Q_{3}^{\text{SRR},i}$
$\displaystyle={-}(\bar{d}_{j}^{\alpha}\sigma_{\mu\nu}P_{R}\,d_{i}^{\beta})\,(\bar{d}_{i}^{\beta}\sigma^{\mu\nu}P_{R}\,d_{i}^{\alpha}),$
(96)

together with the operators

$\displaystyle Q_{25}={Q_{2}^{\text{SRR},i}}$
$\displaystyle=(\bar{d}_{j}^{\alpha}P_{R}\,d_{i}^{\alpha})\,(\bar{d}_{i}^{\beta}P_{R}\,d_{i}^{\beta}),$
$\displaystyle Q_{26}={Q_{4}^{\text{SRR},i}}$
$\displaystyle={-}(\bar{d}_{j}^{\alpha}\sigma_{\mu\nu}P_{R}\,d_{i}^{\alpha})\,(\bar{d}_{i}^{\beta}\sigma^{\mu\nu}P_{R}\,d_{i}^{\beta}),$
(97) and similar for the SRR$,j$ sector $\displaystyle
Q_{27}={Q_{2}^{\text{SRR},j}}$
$\displaystyle=(\bar{d}_{j}^{\alpha}P_{R}\,d_{i}^{\alpha})\,(\bar{d}_{j}^{\beta}P_{R}\,d_{j}^{\beta}),$
$\displaystyle Q_{28}={Q_{4}^{\text{SRR},j}}$
$\displaystyle={-}(\bar{d}_{j}^{\alpha}\sigma_{\mu\nu}P_{R}\,d_{i}^{\alpha})\,(\bar{d}_{j}^{\beta}\sigma^{\mu\nu}P_{R}\,d_{j}^{\beta}),$
(98)

together with

$\displaystyle Q_{1}^{\text{SRR},j}$
$\displaystyle=(\bar{d}_{j}^{\alpha}P_{R}\,d_{i}^{\beta})\,(\bar{d}_{j}^{\beta}P_{R}\,d_{j}^{\alpha}),$
$\displaystyle Q_{3}^{\text{SRR},j}$
$\displaystyle={-}(\bar{d}_{j}^{\alpha}\sigma_{\mu\nu}P_{R}\,d_{i}^{\beta})\,(\bar{d}_{j}^{\beta}\sigma^{\mu\nu}P_{R}\,d_{j}^{\alpha}).$
(99)

Note that we choose the operators $Q_{26}$ and $Q_{28}$ with an opposite sign,
compared to the basis in [11]. For the SRR sector with four different quarks
we define the structures

$\displaystyle Q_{1}^{\text{SRR},Q}$
$\displaystyle=(\bar{d}_{j}^{\alpha}P_{R}\,d_{i}^{\beta})\,(\bar{Q}^{\beta}P_{R}\,Q^{\alpha}),\qquad$
$\displaystyle Q_{3}^{\text{SRR},Q}$
$\displaystyle={-}(\bar{d}_{j}^{\alpha}\sigma_{\mu\nu}P_{R}\,d_{i}^{\beta})\,(\bar{Q}^{\beta}\sigma^{\mu\nu}P_{R}\,Q^{\alpha}),$
(100) $\displaystyle Q_{2}^{\text{SRR},Q}$
$\displaystyle=(\bar{d}_{j}^{\alpha}P_{R}\,d_{i}^{\alpha})\,(\bar{Q}^{\beta}P_{R}\,Q^{\beta}),$
$\displaystyle Q_{4}^{\text{SRR},Q}$
$\displaystyle={-}(\bar{d}_{j}^{\alpha}\sigma_{\mu\nu}P_{R}\,d_{i}^{\alpha})\,(\bar{Q}^{\beta}\sigma^{\mu\nu}P_{R}\,Q^{\beta})\,,$

where the tensor structures have again opposite sign compared to the
convention adopted in [11]. With these definitions we define the operators
$Q_{29}-Q_{40}$

$\displaystyle(Q_{29},\,Q_{30},\,Q_{31},\,Q_{32})$
$\displaystyle=(Q_{1}^{\text{SRR},u},\,Q_{2}^{\text{SRR},u},\,Q_{3}^{\text{SRR},u},\,Q_{4}^{\text{SRR},u}),$
(101) $\displaystyle(Q_{33},\,Q_{34},\,Q_{35},\,Q_{36})$
$\displaystyle=(Q_{1}^{\text{SRR},c},\,\,Q_{2}^{\text{SRR},c},\,\,Q_{3}^{\text{SRR},c},\,\,Q_{4}^{\text{SRR},c}),$
$\displaystyle(Q_{37},\,Q_{38},\,Q_{39},\,Q_{40})$
$\displaystyle=(Q_{1}^{\text{SRR},d_{k}},\,Q_{2}^{\text{SRR},d_{k}},\,Q_{3}^{\text{SRR},d_{k}},\,Q_{4}^{\text{SRR},d_{k}}).$

Finally, as far as the chirality-flipped operators are concerned, their
numbering in the BMU basis is given by

$\displaystyle Q_{40+i}$ $\displaystyle=Q_{i}[P_{L}\leftrightarrow P_{R}],$
(102)

i.e. they are found by interchanging $P_{L}\leftrightarrow P_{R}$ in the “non-
flipped” operators.

##  Appendix B Fierz identities

In the process of transforming from one operator basis to another one requires
the Fierz identities [27] that allow to transfer a given chain of spinors into
another one. We list here the usual Fierz identities valid in $D=4$ dimensions
that we used in our analysis.

All Fierz identities used are of the type $(12)(34)\to(14)(32)$ in which the
exchange of fermion fields $2\leftrightarrow 4$ (or equivalently
$1\leftrightarrow 3$) takes place. In the formulae below $P_{A}$ and $P_{B}$
stand for the usual projectors $P_{L,R}$ but in a given relation
$P_{A}\not=P_{B}$. This means that if $P_{A}=P_{L}$, then $P_{B}=P_{R}$ and
vice versa.

We have then

$(\bar{\psi}_{1}P_{A}\psi_{2})(\bar{\psi}_{3}P_{A}\psi_{4})=-\frac{1}{2}(\bar{\psi}_{1}P_{A}\psi_{4})(\bar{\psi}_{3}P_{A}\psi_{2})-\frac{1}{8}(\bar{\psi}_{1}\sigma_{\mu\nu}P_{A}\psi_{4})(\bar{\psi}_{3}\sigma^{\mu\nu}P_{A}\psi_{2}),$
(103)
$(\bar{\psi}_{1}P_{A}\psi_{2})(\bar{\psi}_{3}P_{B}\psi_{4})=-\frac{1}{2}(\bar{\psi}_{1}\gamma_{\mu}P_{B}\psi_{4})(\bar{\psi}_{3}\gamma^{\mu}P_{A}\psi_{2}),$
(104)
$(\bar{\psi}_{1}\gamma_{\mu}P_{A}\psi_{2})(\bar{\psi}_{3}\gamma^{\mu}P_{A}\psi_{4})=(\bar{\psi}_{1}\gamma_{\mu}P_{A}\psi_{4})(\bar{\psi}_{3}\gamma^{\mu}P_{A}\psi_{2}),$
(105)
$(\bar{\psi}_{1}\gamma_{\mu}P_{A}\psi_{2})(\bar{\psi}_{3}\gamma^{\mu}P_{B}\psi_{4})=-2(\bar{\psi}_{1}P_{B}\psi_{4})(\bar{\psi}_{3}P_{A}\psi_{2}),$
(106)
$(\bar{\psi}_{1}\sigma_{\mu\nu}P_{A}\psi_{2})(\bar{\psi}_{3}\sigma^{\mu\nu}P_{A}\psi_{4})=-6(\bar{\psi}_{1}P_{A}\psi_{4})(\bar{\psi}_{3}P_{A}\psi_{2})+\frac{1}{2}(\bar{\psi}_{1}\sigma_{\mu\nu}P_{A}\psi_{4})(\bar{\psi}_{3}\sigma^{\mu\nu}P_{A}\psi_{2}),$
(107)
$(\bar{\psi}_{1}\sigma_{\mu\nu}P_{A}\psi_{2})(\bar{\psi}_{3}\sigma^{\mu\nu}P_{B}\psi_{4})=0\,.$
(108)

For more Fierz identities involving charge conjugated fields see Appendix A.3
in [1].

Apart from this we also need the color identity

$T^{A}\otimes
T^{A}=\frac{1}{2}(\mathbb{\tilde{1}}-\frac{1}{N_{c}}\mathbb{1}).$ (109)

##  Appendix C Master Formulae for One-loop Operator Insertions

In this section, we present master formulae for the 1-loop operator
insertions. These can be used to obtain the shifts given in Sec.3.1. Consider
a 4-fermion operator

$(\bar{q}_{1}\hat{V}_{1}\Gamma_{1}q_{2})(\bar{q}_{3}\hat{V}_{2}\Gamma_{2}q_{4})\,,$
(110)

here $\hat{V}_{1,2}$ and $\Gamma_{1,2}$ represent the color and Dirac
structures. There are two types of penguin insertions, an _open penguin_ , and
the _closed penguin_. In the next two subsections we evaluate the
corresponding amplitudes.

###  C.1 Open penguin insertion

The open penguin insertion of the operator (110) gives

$P_{op}=W_{\lambda}~{}(-ig_{s}T^{b}\gamma_{\lambda^{\prime}})~{}\left(\frac{-ig^{\lambda{\lambda^{\prime}}}}{q^{2}}\right)\,,$
(111)
$W_{\lambda}=i^{2}\hat{V}_{1}(-ig_{s}T^{a})\hat{V}_{2}I^{\mu\nu}T_{\mu\nu}^{\lambda}\,,$
(112)

where
$T_{\mu\nu}^{\lambda}=\Gamma_{1}\gamma_{\nu}\gamma_{\lambda}\gamma_{\mu}\Gamma_{2}$
and

$I^{\mu\nu}=\int\frac{d^{D}k}{(2\pi)^{D}}\frac{k^{\nu}(k-q)^{\mu}}{k^{2}(k-q)^{2}}=-\frac{i}{16\pi^{2}}\frac{1}{\varepsilon}\left(\frac{1}{6}q_{\mu}q_{\nu}+\frac{1}{12}q^{2}g_{\mu\nu}\right)-\frac{i}{16\pi^{2}}\left(\frac{5}{18}q_{\mu}q_{\nu}+\frac{2}{9}q^{2}g_{\mu\nu}\right).$
(113)

Therefore the finite and infinite parts of $P_{op}$ can be written as

$\displaystyle P_{op}(\rm infinite)$ $\displaystyle=$ $\displaystyle-
C_{op}\frac{\alpha_{s}}{4\pi}\frac{1}{\varepsilon}\left(\frac{1}{6}\frac{q_{\mu}q_{\nu}}{q^{2}}+\frac{1}{12}g_{\mu\nu}\right)\Gamma_{1}\gamma_{\mu}\gamma_{\lambda}\gamma_{\nu}\Gamma_{2}\otimes\gamma^{\lambda}\,,$
(114) $\displaystyle P_{op}(\rm finite)$ $\displaystyle=$ $\displaystyle-
C_{op}\frac{\alpha_{s}}{4\pi}\left(\frac{5}{18}\frac{q_{\mu}q_{\nu}}{q^{2}}+\frac{2}{9}g_{\mu\nu}\right)\Gamma_{1}\gamma_{\mu}\gamma_{\lambda}\gamma_{\nu}\Gamma_{2}\otimes\gamma^{\lambda}.$
(115)

Here $C_{op}=\hat{V}_{1}T^{b}\hat{V}_{2}\otimes T^{b}$.

###  C.2 Closed penguin insertion

The closed penguin insertion of the operator (110) gives

$P_{cl}=W_{\lambda}~{}(-ig_{s}T^{b}\gamma_{\lambda^{\prime}})~{}\left(\frac{-ig^{\lambda{\lambda^{\prime}}}}{q^{2}}\right)\,,$
(116) $W_{\lambda}=(-1)(i^{2})(-ig_{s})~{}{\rm
Tr}(\hat{V}_{1}T^{b})\hat{V}_{2}~{}{\rm
Tr}(\Gamma_{1}\gamma_{\mu}\gamma_{\lambda}\gamma_{\nu})\Gamma_{2}~{}I^{\mu\nu}.$
(117)

Here the $I^{\mu\nu}$ is given by (113).

Therefore the finite and infinite parts of $P_{cl}$ can be written as

$\displaystyle P_{cl}(\rm infinite)$ $\displaystyle=$ $\displaystyle
C_{cl}\frac{\alpha_{s}}{4\pi}\frac{1}{\varepsilon}\left(\frac{1}{6}\frac{q_{\mu}q_{\nu}}{q^{2}}+\frac{1}{12}g_{\mu\nu}\right){\rm
Tr}(\Gamma_{1}\gamma_{\mu}\gamma_{\lambda}\gamma_{\nu})\Gamma_{2}\otimes\gamma^{\lambda}\,,$
(118) $\displaystyle P_{cl}(\rm finite)$ $\displaystyle=$ $\displaystyle
C_{cl}\frac{\alpha_{s}}{4\pi}\left(\frac{5}{18}\frac{q_{\mu}q_{\nu}}{q^{2}}+\frac{2}{9}g_{\mu\nu}\right){\rm
Tr}(\Gamma_{1}\gamma_{\mu}\gamma_{\lambda}\gamma_{\nu})\Gamma_{2}\otimes\gamma^{\lambda}.$
(119)

Here $C_{cl}={\rm Tr}(\hat{V}_{1}T^{b})\hat{V}_{2}\otimes T^{b}$.

###  C.3 Special Cases

In Tables 2 and 3 we give the finite and singular parts for the penguin
operator insertions with various Dirac structures.

Dirac Structure |  $P_{op}$(infinite) |  $P_{op}$(finite)
---|---|---
$\Gamma_{1}=\gamma_{\rho}P_{L}$, $\Gamma_{2}=\gamma^{\rho}P_{L}$ | $C_{op}\frac{\alpha_{s}}{4\pi}\gamma_{\lambda}P_{L}\otimes\gamma^{\lambda}$ | -$C_{op}\frac{\alpha_{s}}{4\pi}\frac{13}{9}\gamma_{\lambda}P_{L}\otimes\gamma^{\lambda}$
$\Gamma_{1}=\gamma_{\rho}P_{L}$, $\Gamma_{2}=\gamma^{\rho}P_{R}$ | 0 | 0
$\Gamma_{1}=P_{L}$, $\Gamma_{2}=P_{L}$ | 0 | 0
$\Gamma_{1}=P_{L}$, $\Gamma_{2}=P_{R}$ | $-C_{op}\frac{\alpha_{s}}{4\pi}\frac{1}{6}\gamma_{\lambda}P_{R}\otimes\gamma^{\lambda}$ | $C_{op}\frac{\alpha_{s}}{4\pi}\frac{13}{8}\gamma_{\lambda}P_{R}\otimes\gamma^{\lambda}$
$\Gamma_{1}=\sigma_{\alpha\beta}P_{L}$, $\Gamma_{2}=\sigma^{\alpha\beta}P_{L}$ | 0 | 0
Table 2: Finite and infinite parts of the open penguin insertion. $C_{op}=\hat{V}_{1}T^{b}\hat{V}_{2}\otimes T^{b}$. Dirac Structure |  $P_{cl}$(finite) |  $P_{cl}$(infinite)
---|---|---
$\Gamma_{1}=\gamma_{\rho}P_{L}$, $\Gamma_{2}=\gamma^{\rho}P_{L}$ | \- $C_{cl}\frac{\alpha_{s}}{4\pi}\frac{13}{9}\gamma_{\lambda}P_{L}\otimes\gamma^{\lambda}$ | $C_{cl}\frac{\alpha_{s}}{4\pi}\frac{1}{3}\gamma_{\lambda}P_{L}\otimes\gamma^{\lambda}$
$\Gamma_{1}=\gamma_{\rho}P_{L}$, $\Gamma_{2}=\gamma^{\rho}P_{R}$ | -$C_{cl}\frac{\alpha_{s}}{4\pi}\frac{13}{9}\gamma_{\lambda}P_{R}\otimes\gamma^{\lambda}$ | $C_{cl}\frac{\alpha_{s}}{4\pi}\frac{1}{3}\gamma_{\lambda}P_{R}\otimes\gamma^{\lambda}$
$\Gamma_{1}=P_{L}$, $\Gamma_{2}=P_{L}$ | 0 | 0
$\Gamma_{1}=P_{L}$, $\Gamma_{2}=P_{R}$ | 0 | 0
$\Gamma_{1}=\sigma_{\alpha\beta}P_{L}$, $\Gamma_{2}=\sigma^{\alpha\beta}P_{L}$ | 0 | 0
Table 3: Finite and infinite parts the of closed penguin insertion.
$C_{cl}={\rm Tr}(\hat{V}_{1}T^{b})\hat{V}_{2}\otimes T^{b}$.

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