# Beyond Neighbourhood-Preserving Transformations for Quantization-Based
Unsupervised Hashing

Sobhan Hemati H.R. Tizhoosh<EMAIL_ADDRESS>Kimia Lab, University of
Waterloo, Waterloo, ON, Canada  Vector Institute, MaRS Centre, Toronto, ON,
Canada

###### Abstract

An effective unsupervised hashing algorithm leads to compact binary codes
preserving the neighborhood structure of data as much as possible. One of the
most established schemes for unsupervised hashing is to reduce the
dimensionality of data and then find a rigid (neighborhood-preserving)
transformation that reduces the quantization error. Although employing rigid
transformations is effective, we may not reduce quantization loss to the
ultimate limits. As well, reducing dimensionality and quantization loss in two
separate steps seems to be sub-optimal. Motivated by these shortcomings, we
propose to employ both rigid and non-rigid transformations to reduce
quantization error and dimensionality simultaneously. We relax the
orthogonality constraint on the projection in a PCA-formulation and regularize
this by a quantization term. We show that both the non-rigid projection matrix
and rotation matrix contribute towards minimizing quantization loss but in
different ways. A scalable nested coordinate descent approach is proposed to
optimize this mixed-integer optimization problem. We evaluate the proposed
method on five public benchmark datasets providing almost half a million
images. Comparative results indicate that the proposed method mostly
outperforms state-of-art linear methods and competes with end-to-end deep
solutions.

###### keywords:

###### MSC:

41A05, 41A10, 65D05, 65D17 Keyword1, Keyword2, Keyword3

##  1 Introduction

Search in big image data is one of the major challenges that has many
applications in retrieval, classification and clustering of digital images. A
well-established approach toward indexing and searching in mass repositories
is hashing that assigns binary codes to data records for fast and efficient
access. Hashing has been applied to different problems where fast nearest
neighbour search is necessary e.g, large-scale clustering Yunchao Gong et al.
(2015).

(a) ITQ

(b) SNRQ

Fig. 1: Histogram of transformed MNIST dataset using a) ITQ (rigid
transformation) and b) SNRQ (rigid and non-rigid transformations).

Hashing algorithms can be categorized in two main sub-groups: data-independent
and data-dependent algorithms Wang et al. (2018). The locality sensitive
hashing (LSH) Andoni and Indyk (2006) is one of the first representatives in
the first group where hash functions are constructed using random projections.
Although these methods are faster compared with data-dependent algorithms,
their performance is inferior for compact binary codes Paulevé et al. (2010).
To address this shortcoming, many data-dependent (or learning-based) hashing
algorithms which are divided into supervised and unsupervised methods have
been proposed. Supervised hashing methods incorporate label information to
learn similarity-preserving binary codes while in unsupervised hashing, which
is the focus of this paper the affinity information in the data is used for
code learning Weiss et al. (2009).

The main challenge in learning similarity-preserving binary codes is the
discrete nature of the optimization problems. To tackle this problem different
approaches have been proposed including minimizing quantization loss in
Iterative Quantization (ITQ) Gong et al. (2013), Angular Quantization (AQ)
Gong et al. (2012), equalizing variance across projections in isotropic
hashing (IsoHash) Kong and Li (2012), reconstruction loss in an encoder-
decoder architecture like in binary autoencoder (BA) Carreira-Perpinan and
Raziperchikolaei (2015), and direct binary code learning in spectral hashing
(SH) Weiss et al. (2009) and its sparse version Shao et al. (2012). and more
recent developments including Anchor Graph Hashing (AGH) Liu et al. (2011),
and Discrete Graph Hashing (DGH) Liu et al. (2014) Large Graph Hashing with
Spectral Rotation (LGHSR) Li et al. (2017), Discrete Spectral Hashing (DSH) Hu
et al. (2018), and Efficient Spectral Hashing (ESH) Hemati et al. (2020).
Other recent works on unsupervised hashing include, Optimal Projection Hashing
(OPH) Chu et al. (2019) $k$-Nearest Neighbors Hashing (KNNH) He et al. (2019),
and Simultaneous Compression and Quantization (SCQ) Hoang et al. (2020).
Although deep learning has been mainly applied to the supervised hashing
problem, some unsupervised deep hashing algorithms have recently been
proposed. Some recent deep unsupervised hashing algorithms include deep
hashing (DH) Liong et al. (2015), Unsupervised Hashing with Binary Deep Neural
Network (UHBDNN) Do et al. (2016), Deepbit Lin et al. (2016), and Similarity-
Adaptive Deep Hashing (SADH) Shen et al. (2018).

Employing a neighbourhood preserving transformation used in many hashing
algorithms. For example in the ITQ, IsoHash, AQ, and OPH a rotation matrix is
employed to refine projections. Besides, neighbour preserving transformation
has also been used in LGHSR and SCQ to improve the quality of SH relaxed
solution and simultaneous projection and quantization of data respectively.

The main issue with binarization is destroying neighbourhood structure of
data. Having this in mind, rotation matrices seem to be a good choice for
reducing quantization loss as they preserve neighbourhood. Although they are
effective, we argue that a single rotation is not powerful enough to minimize
the quantization error. Instead, we propose to employ transformations beyond
rotation that even corrupt neighbourhood structure of data in favour pushing
for quantization. As we will see, such transformation coupled with a rotation
leads to high quality binary codes that outperforms state-of-art linear
unsupervised hashing methods. Although the proposed idea can be applied to
many hashing algorithms that employ a single rotation for reducing
quantization error, here we choose to develop this idea on top of ITQ which is
very fast and still among competitive hashing methods. In ITQ, the data is
projected to lower dimensionality and then the optimal rotation is found such
that it minimizes the quantization error in two separate steps.

In our method, we employ a transformation beyond rotation and also resolve the
discontinuity in projecting data to lower dimensionality and reduce
quantization error. We formulate and optimize our objective function such that
in our algorithm:

  1. 1.

The data is projected and quantized simultaneously. This means unlike many
conventional linear unsupervised hashing methods, the projection matrix is
contributed towards reducing quantization error.

  2. 2.

We relax the orthogonality on projection matrix in favor of reducing
quantization error even if it corrupts the original neighbourhood. We
experiment different matrix norms for achieving this non-rigid transformation.

  3. 3.

An efficient sequential update scheme is proposed for learning the projection
matrix.

The left plot in Fig. 1 shows the histogram of transformed data for a
neighbourhood preserving transformation (ITQ) while the right shows the same
data transformed by a non-neighbourhood preserving operations using our
proposed method Sequential Non-rigid Quantization (SNRQ). Apparently, SNRQ
pushes data points more distinctively toward +1 and -1 modes compared with
ITQ.

Using extensive quantitative experiments on five public detest and also
qualitative results, we show that although our transformation corrupts the
neighbourhood of data, the final binary codes obtained using our method,
preserve more neighbourhood compared with many other linear hashing methods.

##  2 Proposed Method

Formulation –  In this section, we formulate an optimization problem that
follows two objectives. Firstly, it is desirable to jointly learn a projection
matrix and minimize the quantization loss of projected data. Secondly, the
orthogonality constraint on the projection matrix is relaxed such that we can
employ a non-rigid transformation for reducing quantization error. These two
objectives together enable non-orthogonal projection matrix to contribute to
minimizing quantization loss in a way different from the orthogonal rotation
matrix. The first objective is achieved by joining the two well known ITQ
steps, namely applying PCA to the data and minimizing quantization loss. Let
$\mathbf{X}\in\rm I\\!R^{n\times D}$ represents zero-centered data in which
$n$ is the number of training data points and $D$ is the data dimensionality.
It is well understood that the projection matrix $\mathbf{W}\in\rm
I\\!R^{D\times K}$ or $K$ principal components of data can be obtained by
maximizing the objective function

$\underset{\mathbf{W}}{\text{arg
max}}\quad\textrm{\emph{Tr}}\\{\mathbf{W}^{T}\mathbf{X}^{T}\mathbf{X}\mathbf{W}\\}\quad\textrm{s.t.}\quad\mathbf{W}^{T}\mathbf{W}=\mathbf{I}_{K},$
(1)

where $\mathbf{I}_{K}$ is the $K\times K$ identity matrix and _Tr{}_ is the
trace of a matrix. Here we define $\mathbf{V}$ as
$\mathbf{V}=\mathbf{X}\mathbf{W}$ with $\mathbf{V}\in\rm I\\!R^{n\times K}.$
We would like to minimize the quantization loss of transformed data. To do
this, we find an orthogonality relaxed matrix $\mathbf{W}$ (the relaxed
orthogonality constraint on $\mathbf{W}$ will be formulated later in Eq. 3)
and an $K\times K$ orthogonal rotation matrix $\mathbf{R}$ such that the
quantization loss of thresholding the transformed data
$\mathbf{V}\mathbf{R}=\mathbf{X}\mathbf{W}\mathbf{R}$ at zero is minimized.
This can be formulated as minimizing the following:

$Q(\mathbf{B},\mathbf{R},\mathbf{W})=\|\mathbf{X}\mathbf{W}\mathbf{R}-\mathbf{B}\|^{2}_{F}\quad\textrm{s.t.}\quad\mathbf{R}^{T}\mathbf{R}=\mathbf{I}_{K},$
(2)

where $\mathbf{B}\in\\{-1,1\\}^{n\times k}$ is the corresponding binary
representation of $\mathbf{X}$. In order to jointly learn the projection
matrix $\mathbf{W}$ and minimize the quantization loss using both non-rigid
$\mathbf{W}$ and rigid $\mathbf{R}$, we relax the orthogonality constrain on
the projection in PCA formulation and also regularize this by a quantization
term which leads to our proposed objective function:

$\begin{split}\underset{\mathbf{W},\mathbf{R},\mathbf{B}}{\text{arg
max}}\quad\emph{J}(\mathbf{W},\mathbf{R},\mathbf{B})\\!=\\!\textrm{\emph{Tr}}\\{\mathbf{W}^{T}\\!\mathbf{C_{x}}\\!\mathbf{W}\\}-\\\
\alpha\|\mathbf{X}\mathbf{W}\mathbf{R}\\!-\\!\mathbf{B}\\!\|^{2}_{F}-\beta\|\mathbf{W}^{T}\mathbf{W}\\!-\\!\mathbf{I}_{K}\|^{2}_{F},\end{split}$
(3)

subject to $\mathbf{R}^{T}\mathbf{R}=\mathbf{I}_{K}$ and
$\mathbf{B}\in\\{-1,1\\}^{n\times k}$ where $\alpha$ and $\beta$ are
quantization and rigidness regularization parameters, and finally
$\mathbf{C_{x}}=\mathbf{X}^{T}\mathbf{X}$. Note that in Eq. 3 both
$\mathbf{W}$ and $\mathbf{R}$ are contributing to minimizing quantization
loss, $\|\mathbf{X}\mathbf{W}\mathbf{R}-\\!\mathbf{B}\\!\|^{2}_{F}$, but in
different ways as orthogonality constraint on $\mathbf{W}$ is smoothed whereas
$\mathbf{R}$ is a pure orthogonal rotation matrix. This is one of the main
differences between the proposed method and other existing methods e.g., in
contrast to ITQ, IsoHash, AQ, and SCQ.

Optimization –  In order to jointly minimize the quantization loss and learn
the projection matrix, we need to find $\mathbf{W}$, $\mathbf{R}$, and
$\mathbf{B}$ in a way to maximize the objective function in Eq. 3. Due to
binary constraints on $\mathbf{B}$, the optimization is intractable. Hence,
inspired by technique used in ITQ paper, we use a coordinate descent approach
to optimize the objective function over $\mathbf{W}$, $\mathbf{R}$, and
$\mathbf{B}$. To this end, in each step we consider one of the $\mathbf{W}$,
$\mathbf{R}$, and $\mathbf{B}$ variable and the two other matrices are assumed
to be constant.

Fix $\mathbf{R}$ and $\mathbf{B}$, and update $\mathbf{W}$: In order to
optimize the objective in Eq. 3 with respect to $\mathbf{W}$, we easily
calculate the derivative with respect to $\mathbf{W}$ as follow:

$\begin{split}\frac{\partial\emph{J}(\mathbf{W},\mathbf{R},\mathbf{B})}{\partial\mathbf{W}}=2\mathbf{\mathbf{C_{x}}}\mathbf{W}-\alpha\left(2\mathbf{X}^{T}(\mathbf{X}\mathbf{W}\mathbf{R}-\mathbf{B})\mathbf{R}^{T}\right)\\\
-\beta\left(4\mathbf{W}(\mathbf{W}^{T}\mathbf{W}-\mathbf{I}_{K})\right).\end{split}$
(4)

Having the gradient w.r.t $\mathbf{W}$, we found that the L-BFGS-B optimizer
Zhu et al. (1997) obtain a good solution for $\mathbf{W}$. However, as each
variable should be updated multiple times (while other are fixed) we observed
that optimizing for $\mathbf{W}$ directly is a time-consuming procedure. To
mitigate this challenge, in the following we propose a sequential approach for
updating $\mathbf{W}$. As we will show in experiments, sequential approach
achieves comparable performance while reducing training time significantly. In
sequential scheme, each time one column of $\mathbf{W}$ is updated while the
rest of columns are considered fixed. First, let’s expand the second term
(quantization loss) in Eq. 3:

$\displaystyle\|\mathbf{X}\mathbf{W}\mathbf{R}-\mathbf{B}\|^{2}_{F}=\|\mathbf{X}\mathbf{W}\mathbf{R}\|^{2}_{F}-2\textrm{\emph{Tr}}\\{\mathbf{X}\mathbf{W}\mathbf{R}\mathbf{B}^{T}\\}+\|\mathbf{B}\|^{2}_{F}$
(5) $\displaystyle=$
$\displaystyle\textrm{\emph{Tr}}\\{\mathbf{W}^{T}\mathbf{C_{x}}\mathbf{W}\\}-2\textrm{\emph{Tr}}\\{\mathbf{W}(\mathbf{R}\mathbf{B}^{T}\mathbf{X})\\}+const.$

Now let’s expand the relaxed orthogonality constraint (the third term in Eq.
3):

$\displaystyle\|\mathbf{W}^{T}\mathbf{W}-\mathbf{I}_{K}\|^{2}_{F}=\|\mathbf{W}^{T}\mathbf{W}\|^{2}_{F}-2\textrm{\emph{Tr}}\\{\mathbf{W}^{T}\mathbf{W}\mathbf{I}_{K}\\}+const$
(6) $\displaystyle=$
$\displaystyle\textrm{\emph{Tr}}\\{(\mathbf{W}\mathbf{W}^{T})(\mathbf{W}\mathbf{W}^{T})\\}-2\textrm{\emph{Tr}}\\{\mathbf{W}\mathbf{W}^{T}\\}+const.$

Using Eqs. 5 and 6 for the second and third terms in Eq. 3, respectively, and
defining $\mathbf{C_{y}}=\mathbf{R}\mathbf{B}^{T}\mathbf{X}$ provides us with
the objective function for the case that $\mathbf{W}$ is variable:

$\displaystyle\underset{\mathbf{W}}{\text{arg
max}}\quad(1-\alpha)\textrm{\emph{Tr}}\\{\mathbf{W}^{T}\mathbf{C_{x}}\mathbf{W}\\}+2\alpha\textrm{\emph{Tr}}\\{\mathbf{W}\mathbf{C_{y}}\\}$
(7)
$\displaystyle-\beta\textrm{\emph{Tr}}\\{(\mathbf{W}\mathbf{W}^{T})(\mathbf{W}\mathbf{W}^{T})\\}+2\beta\textrm{\emph{Tr}}\\{\mathbf{W}\mathbf{W}^{T}\\}.$

Let’s start formulating the problem such that one column of $\mathbf{W}$ is
considered variable while the remaining columns are constant. To this end, the
$k$-th column of $\mathbf{W}$ that is considered variable is denoted by
$\mathbf{z}_{k}$ and all other columns are shown by $\mathbf{W^{\prime}}$. In
this case, for the first term we receive

$\displaystyle\textrm{\emph{Tr}}\\{\mathbf{W}^{T}\mathbf{C_{x}}\mathbf{W}\\}=\textrm{\emph{Tr}}\\{\mathbf{C_{x}}\mathbf{W}\mathbf{W}^{T}\\}=$
(8)
$\displaystyle\textrm{\emph{Tr}}\\{\mathbf{C_{x}}(\mathbf{W^{\prime}}\mathbf{W^{\prime}}^{T}+\mathbf{z}_{k}\mathbf{z}_{k}^{T})\\}=const+\mathbf{z}_{k}^{T}\mathbf{C_{x}}\mathbf{z}_{k}.$

Note that here first we used the fact that
$\textrm{\emph{Tr}}\\{\mathbf{C_{x}}\mathbf{z}_{k}\mathbf{z}_{k}^{T}\\}=\textrm{\emph{Tr}}\\{\mathbf{z}_{k}^{T}\mathbf{C_{x}}\mathbf{z}_{k}\\}$
and then removed $\textrm{\emph{Tr}}\\{\cdot\\}$ as the
$\mathbf{z}_{k}^{T}\mathbf{C_{x}}\mathbf{z}_{k}$ is scalar. Similarly, if we
define the $k$-th row of $\mathbf{C_{y}}$ as $\mathbf{u}_{k}^{T}$ and the rest
of the rows as $\mathbf{C^{\prime}_{y}}$, then the second term can be written
as

$\displaystyle\textrm{\emph{Tr}}\\{\mathbf{W}\mathbf{C_{y}}\\}\\!=\\!\textrm{\emph{Tr}}\\{\mathbf{W^{\prime}}\mathbf{C^{\prime}_{y}}+\mathbf{z}_{k}\mathbf{u}_{k}^{T}\\}\\!=$
(9) $\displaystyle
const+\\!\textrm{\emph{Tr}}\\{\mathbf{u}_{k}^{T}\mathbf{z}_{k}\\}\\!=const+\\!\mathbf{u}_{k}^{T}\mathbf{z}_{k}.$

For the third term in Eq. 7 we write

$\displaystyle\textrm{\emph{Tr}}\\{(\mathbf{W}\mathbf{W}^{T})(\mathbf{W}\mathbf{W}^{T})\\}=$
(10)
$\displaystyle\textrm{\emph{Tr}}\\{(\mathbf{W^{\prime}}\mathbf{W^{\prime}}^{T}+\mathbf{z}_{k}\mathbf{z}_{k}^{T})(\mathbf{W^{\prime}}\mathbf{W^{\prime}}^{T}+\mathbf{z}_{k}\mathbf{z}_{k}^{T})\\}=$
$\displaystyle
const+2\mathbf{z}_{k}^{T}\mathbf{W^{\prime}}\mathbf{W^{\prime}}^{T}\mathbf{z}_{k}+\mathbf{z}_{k}^{T}\mathbf{z}_{k}\mathbf{z}_{k}^{T}\mathbf{z}_{k}.$

As
$\mathbf{z}_{k}^{T}\mathbf{W^{\prime}}\mathbf{W^{\prime}}^{T}\mathbf{z}_{k}$
and $\mathbf{z}_{k}^{T}\mathbf{z}_{k}\mathbf{z}_{k}^{T}\mathbf{z}_{k}$ are
scalars, $\textrm{\emph{Tr}}\\{\cdot\\}$ can be removed. Similarly, for the
last term we obtain

$\displaystyle\textrm{\emph{Tr}}\\{\mathbf{W}\mathbf{W}^{T}\\}\\!=\\!\textrm{\emph{Tr}}\\{\mathbf{W^{\prime}}\mathbf{W^{\prime}}^{T}+\mathbf{z}_{k}\mathbf{z}_{k}^{T}\\}\\!=$
(11) $\displaystyle
const+\\!\textrm{\emph{Tr}}\\{\mathbf{z}_{k}^{T}\mathbf{z}_{k}\\}\\!=\\!const+\mathbf{z}_{k}^{T}\mathbf{z}_{k}.$

Incorporating Eqs. 8, 9, 10, and 11 into Eq. 7, if we set

$\mathbf{Q}=(1-\alpha)\mathbf{C_{x}}-2\beta\mathbf{W^{\prime}}\mathbf{W^{\prime}}^{T}+2\beta\mathbf{I}_{D},$
(12)

then by maximizing the following objective function one can obtain the
$\mathbf{z}_{k}$, $k$-th column of $\mathbf{W}$:

$\underset{\mathbf{z}_{k}}{\text{arg
max}}\quad\emph{J}(\mathbf{z}_{k})\\!=\\!\mathbf{z}_{k}^{T}\mathbf{Q}\mathbf{z}_{k}\\!+\\!2\alpha\mathbf{u}_{k}^{T}\mathbf{z}_{k}\\!-\\!\beta\mathbf{z}_{k}^{T}\mathbf{z}_{k}\mathbf{z}_{k}^{T}\mathbf{z}_{k}.$
(13)

To update each column of $\mathbf{W}$, i.e, $\mathbf{z}_{k}$, where $k=1\dots
K,$ we have to find $\mathbf{z}_{k}$ that maximizes Eq. 13. The gradient of
objective function (Eq. 13) can be calculated as

$\frac{\partial\emph{J}(\mathbf{z}_{k})}{\partial\mathbf{z}_{k}}=2\mathbf{Q}\mathbf{z}_{k}+2\alpha\mathbf{u}_{k}-4\beta\mathbf{z}_{k}(\mathbf{z}_{k}\mathbf{z}_{k}^{T}).$
(14)

Now, we use the L-BFGS-B optimizer to obtain solutions. As we will show in
experiments, this sequential learning scheme significantly reduces the
training time of the proposed algorithm without sacrificing the quality of
binary codes. This efficiency is in part because of simplifying the objective
function and also reducing the search space by dividing the problem to $K$
sub-problems.

Fix $\mathbf{R}$ and $\mathbf{W}$, and update $\mathbf{B}$: In this case, the
optimization in Eq. 3 takes the following form:

$\displaystyle\underset{\mathbf{B}}{\text{arg min}}$
$\displaystyle\|\mathbf{X}\mathbf{W}\mathbf{R}-\mathbf{B}\|^{2}_{F}\quad\textrm{s.t.}\quad\mathbf{B}\in\\{-1,1\\}^{n\times
k}.$ (15)

Having in mind that $\mathbf{R}$ and $\mathbf{W}$ are fixed, clearly,
minimization of Eq. 15 with respect to $\mathbf{B}$ is equivalent the
maximization of
$\textrm{\emph{Tr}}\\{(\mathbf{X}\mathbf{W}\mathbf{R})^{T}\mathbf{B}\\}$ where
elements of $\mathbf{B}$ can be either 1 or -1. As we know from ITQ Gong et
al. (2013), the optimal $\mathbf{B}$ can be calculated as

$\mathbf{B}=sgn(\mathbf{V}\mathbf{R})=sgn(\mathbf{X}\mathbf{W}\mathbf{R}).$
(16)

Fix $\mathbf{W}$ and $\mathbf{B}$, and update $\mathbf{R}$: For the case that
$\mathbf{R}$ is variable, maximizing Eq. 3 is equivalent to

$\displaystyle\underset{\mathbf{R}}{\text{arg min}}$
$\displaystyle\|\mathbf{V}\mathbf{R}-\mathbf{B}\|^{2}_{F}\quad\textrm{s.t.}\quad\mathbf{R}^{T}\mathbf{R}=\mathbf{I}_{K}.$
(17)

This is the Orthogonal Procrustes problem where a rotation matrix is found
such that two point sets are aligned with each other. Here, these two point
sets are the target binary code matrix $\mathbf{B}$ and projected data
$\mathbf{V}$. This problem has closed form solution when $\mathbf{R}$ is a
square orthogonal (rotation) matrix Schönemann (1966) which is

$\mathbf{R}=\hat{{\mathbf{S}}}{\mathbf{S}}^{T}.$ (18)

where ${\mathbf{S}}\mathbf{\Lambda}\hat{{\mathbf{S}}}^{T}$ is the singular
value decomposition (SVD) of the $K$ by $K$ matrix $\mathbf{B}^{T}\mathbf{V}$.

In summary, in each iteration, Eq. 16 and Eq. 18 are used to update
$\mathbf{B}$ and $\mathbf{R}$, respectively.To update $\mathbf{W}$, in each
iteration, $K$ optimization problems for $K$ columns of $\mathbf{W}$ are
solved. After obtaining $\mathbf{R}$, and $\mathbf{W}$, the binary
representation of the data $\mathbf{X}$ can be obtained
$\mathbf{B}=sgn(\mathbf{V}\mathbf{R})=sgn(\mathbf{X}\mathbf{W}\mathbf{R})$.
When the direct optimization is used for updating $\mathbf{W}$ we denote our
method Non-rigid Quantization (NRQ) and for the sequential case this denoted
by SNRQ. Algorithm 1 summarizes the proposed scheme to update all three
matrices method.

Implementation Note –  We use $K$ eigenvectors of $\mathbf{C}_{x}$
corresponding to $K$ largest eigenvalues as initialization of $\mathbf{W}$.
For $\mathbf{R}$, we use the rotation matrix obtained by ITQ as initialization
of $\mathbf{R}$. Although we will show the algorithm is robust to different
values for $\alpha$ and $\beta$, to avoid tuning regularization parameters, we
set $\alpha=3$ and $\beta=0.01$ for all datasets and experiments. In order to
choose $\alpha$, the only consideration is to keep it larger than 1. This can
be understood based on Eq. 12 which shows there is a trade-off between
reducing quantization error and maximizing variance across projections. This
is clear that the quantization error is our priority. The algorithm is fairly
robust to $\beta$, and based on our experiments, any number between $0.01$ to
$50$ easily does the job. For the number of iterations, experiments on a
variety of detests show that 70 iterations are generally sufficient, and after
that, the loss function does not change significantly. As a results, We set
the number of iterations $N$ to 70. An implementation of the proposed method
is provided in supplementary material.

Algorithm 1 The Proposed SNRQ Algorithm

Input: Data matrix X, number of iterations $N$
regularization parameters $\alpha$ and $\beta$
Output: Projection matrix $\mathbf{W}$, rotation matrix $\mathbf{R}$
Initialization: Initialize $\mathbf{W}$ and $\mathbf{R}$

1:for iteration$=1,2,\ldots,N$ do

2: $\mathbf{V}\leftarrow\mathbf{X}\mathbf{W}$

3: $\mathbf{B}\leftarrow sgn(\mathbf{V}\mathbf{R})$

4:
$\textbf{S}\Lambda\hat{{\textbf{S}}}^{T}\leftarrow\textrm{SVD}(\textbf{B}^{T}\mathbf{V})$

5: $\textbf{R}\leftarrow\hat{{\textbf{S}}}{\textbf{S}}^{T}$

6: $\mathbf{C_{y}}\leftarrow\mathbf{R}\mathbf{B}^{T}\mathbf{X}$

7: for $k=1,2,\ldots,K$ do

8: $\mathbf{u}_{k}^{T}\leftarrow\mathbf{C_{y}}[k,:]$

9:
$\mathbf{Q}\leftarrow(1-\alpha)\mathbf{C_{x}}-2\beta\mathbf{W^{\prime}}\mathbf{W^{\prime}}^{T}+2\beta\mathbf{I}_{K}$

10: Solve Eq. 13 for each column of $\mathbf{W}$, i.e., $\mathbf{z}_{k}$

11:
$\emph{J}(\mathbf{z}_{k})=\mathbf{z}_{k}^{T}\mathbf{Q}\mathbf{z}_{k}+2\alpha\mathbf{u}_{k}^{T}\mathbf{z}_{k}-\beta\mathbf{z}_{k}^{T}\mathbf{z}_{k}\mathbf{z}_{k}^{T}\mathbf{z}_{k}$

12:
$\frac{\partial\emph{J}(\mathbf{z}_{k})}{\partial\mathbf{z}_{k}}=2\mathbf{Q}\mathbf{z}_{k}+2\alpha\mathbf{u}_{k}-4\beta\mathbf{z}_{k}(\mathbf{z}_{k}\mathbf{z}_{k}^{T})$

13: Update the $k$-th column of $\mathbf{W}$, i.e., $\mathbf{z}_{k}$

14: $\mathbf{W}[:,k]\leftarrow\mathbf{z}_{k}$

15: end for

16:end for

##  3 Experiments and Results

Datasets and Evaluation protocol The performance of the proposed SNRQ
algorithm is evaluated on five standard benchmark datasets, MNIST LeCun et al.
(1998), CIFAR-10 Krizhevsky (2009), an unbalanced dataset LabelMe-12-50K Uetz
and Behnke (2009), a medical image dataset NCT-CRC-HE-100K Macenko et al.
(2009), and a multi-label NUS-WIDE Chua et al. (2009). These datasets provide
a total of 489,000 images for learning and testing.

The MNIST dataset contains 70,000 gray-scale images all of size $28\times 28$
pixels. There are 10 classes in this dataset for handwritten digits.

The CIFAR-10 dataset is a 10-class dataset consisting of 60,000 color images
of size $32\times 32$ pixels.

The LabelMe-12-50K dataset is a 12-class dataset containing 50,000 images of
size $256\times 256$ pixels. This dataset is highly imbalanced such that five
classes constitute 91% of all images while there is one class that only
contains 0.6% of the samples. The images of this dataset have multiple label
values between zero and one. In our experiments, same as previous works that
employed this dataset He et al. (2019) for evaluating hashing algorithms, we
choose the class of the largest label value as the image label.

The NCT-CRC-HE-100K dataset is 9-class histopathology dataset containing
100,000 non-overlapping image patches from hematoxylin & eosin stained (H&E)
images of human colorectal cancer and normal tissue. All images are $224\times
224$ pixels and color-normalized.

The NUS-WIDE dataset is a multilabel dataset that contains 269,000 images
collected from Flickr. This database contains 81 ground-truth concepts.

Hash Code Evaluation To evaluate the performance of the SNRQ, we use standard
measures for image retrieval quality assessment. These measures include mean
Average Precision (mAP), and precision at $M$ samples (e.g., precision@1000).
Briefly, mAP measures the overall performance of the retrieval over all
classes, whereas precision@$M$ calculates the proportion of true positive over
top $M$ retrieved samples.

Table 1: Comparison of retrieval performance based on mAP and precision@1000 on MNIST dataset represented by 512-D GIST descriptor. The best performance is highlighted in boldface. KMH: K-means Hashing He et al. (2013), SpH: spherical hashing Heo et al. (2012). | mAP % | precision % @1000
---|---|---
Method | 16 bits | 32 bits | 64 bits | 16 bits | 32 bits | 64 bits
SH | 32.59 | 33.23 | 30.65 | - | - | -
SpH | 31.27 | 36.80 | 41.40 | - | - | -
KMH | 31.96 | 37.39 | 41.11 | - | - | -
BA | 48.48 | 51.72 | 52.73 | - | - | \- -
ITQ | 46.37 | 50.59 | 53.69 | 69.67 | 75.13 | 80.45
KNNH | 53.07 | 61.11 | 65.55 | 73.99 | 83.32 | 87.05
SCQ | 62.39 | 74.49 | 72.23 | 79.26 | 88.46 | 88.90
NRQ (Ours) | 71.46 | 69.44 | 72.79 | 84.59 | 85.49 | 87.30
SNRQ (Ours) | 64.70 | 76.98 | 73.48 | 80.00 | 88.53 | 88.27
Table 2: Comparison of retrieval performance, based on mAP, for 16, 32 and 64 bits for CIFAR-10 and macro mAP (average over classes) for LabelMe-12-50k datasets both represented by 4096-D VGG-FC7 descriptors. | CIFAR-10 | LabelMe-12-50k
---|---|---
Method | 16 bits | 32 bits | 64 bits | 16 bits | 32 bits | 64 bits
SH | 18.31 | 16.54 | 15.78 | 12.60 | 12.59 | 12.24
SpH | 18.82 | 20.93 | 23.40 | 13.59 | 15.10 | 17.03
KMH | 18.68 | 20.82 | 22.87 | 13.36 | 15.47 | 16.58
BA | 25.38 | 26.16 | 27.99 | 16.96 | 18.42 | 20.80
ITQ | 26.82 | 27.38 | 28.73 | 18.06 | 19.40 | 20.73
DGH | 27.73 | 27.44 | 28.01 | 21.45 | 22.74 | 25.41
LGHSR | 27.83 | 25.87 | 22.12 | 21.10 | 23.49 | 23.98
DSH | 27.72 | 25.36 | 22.12 | 24.70 | 23.78 | 24.35
SCQ | 27.52 | 27.42 | 30.34 | 22.89 | 24.95 | 26.50
KNNH | 29.06 | 30.82 | 32.60 | 20.13 | 23.79 | 26.22
NRQ (Ours) | 31.42 | 30.87 | 33.05 | 24.27 | 27.40 | 28.65
SNRQ (Ours) | 30.10 | 31.82 | 33.10 | 25.14 | 26.80 | 28.50

Results on MNIST dataset:

Following the setting of He et al. (2019) for MNIST data, each image is
presented by a GIST 512-D descriptor, 10% of each class is considered for
query set and the remaining data is used as training set. Table 1 shows the
results for MNIST in terms of mAP, and precision@1000. Clearly, the NRQ and
SNRQ methods provide an improvement over other methods in all cases with only
one exception for precision@1000 where the SCQ slightly outperforms SNRQ with
less than 0.5% in 64 bit.

Results on CIFAR-10 dataset: For this dataset, following the common setting He
et al. (2019); Zhang et al. (2015), we used deep 4096-D features extracted
from VGG network Simonyan and Zisserman (2014) and sample 10% of each class as
query set and the remaining instances as training set. The left half of Table
2 represents the results for this experiment setting. As it can be seen, NRQ
and SNRQ outperform state-of-art namely KNNH He et al. (2019) and SCQ Hoang et
al. (2020).

Results on LabelMe-12-50k dataset: As it was pointed out, this dataset is
highly imbalanced. To ascertain a fair comparison, we calculate mAP values
that are macro averages over all classes. Following the common setting He et
al. (2019) we sample 10% of each class to construct the query set, and the
remaining data points as training set. Besides, VGG network has been used to
extract feature vectors. As Table 2 shows, NRQ and SNRQ are performing better
and outperform state-of-art methods with a large gap.

Results on NCT-CRC-HE-100K dataset: For this dataset, we used EfficientNet Tan
and Le (2019) pre-trained on ImageNet to extract 1280-D feature vectors from
images. We randomly sampled 70,000 images (out of 100K) for training set and
the rest of images (30,000) for test. For efficiency, we reduced the feature
vector dimensionality to 512 values by the PCA. Table 3 shows that NRQ and
SNRQ outperform competitive methods based on mAP and precision@1000 with a
significant gap.

Table 3: Comparison of retrieval performance based on mAP, and precision@1000 on NCT-CRC-HE-100K dataset represented by features extracted by EfficientNet. | mAP % | precision % @1000
---|---|---
Method | 16 bits | 32 bits | 64 bits | 16 bits | 32 bits | 64 bits
ITQ | 55.8 | 57.8 | 59.5 | 68.8 | 72.6 | 75.4
DGH | 49.85 | 47.24 | 51.61 | 74.08 | 77.14 | 77.05
LGHSR | 55.28 | 47.74 | 38.65 | 75.66 | 78.21 | 77.13
DSH | 58.75 | 45.63 | 37.27 | 76.70 | 78.71 | 77.73
KNNH | 58.02 | 61.47 | 64.28 | 70.27 | 74.91 | 78.16
SCQ | 64.90 | 67.25 | 67.75 | 76.03 | 80.07 | 81.01
NRQ (Ours) | 75.32 | 73.08 | 77.25 | 83.94 | 83.32 | 86.91
SNRQ (Ours) | 68.5 | 75.4 | 78.7 | 80.1 | 84.5 | 87.0

Results on NUS-WIDE dataset: For NUS-WIDE experiments, we used the common
setup in many hashing papers Liu et al. (2011); Shen et al. (2018). In this
setting, images are selected which their labels are among the 21 most frequent
labels. This leads to 195,834 images. We randomly sample 2,100 images (100
images from each class) from 195,834 images for the test set and the rest of
the images are used for training the hash function and populating the hash
table. We used VGG-F network Chatfield et al. (2014) to extract features from
images. Results for this experiment are reported in Table 4. The results for
other algorithms mainly are directly reported from literature. Shen et al.
(2018).

Table 4: Comparison of retrieval performance on NUS-WIDE dataset represented by VGG-F deep features. | mAP % | precision % @5000
---|---|---
Method | 16 bits | 32 bits | 64 bits | 16 bits | 32 bits | 64 bits
SH | 44.74 | 42.60 | 42.36 | 59.15 | 54.98 | 54.18
AGH | 49.80 | 47.14 | 44.72 | 70.43 | 70.29 | 69.29
DGH | 54.03 | 52.74 | 49.64 | 71.15 | 71.91 | 70.66
LGHSR | 50.79 | 47.72 | 45.45 | 66.79 | 68.59 | 67.60
ITQ | 53.39 | 54.61 | 55.75 | 65.80 | 68.69 | 70.59
DSH | 49.87 | 47.07 | 45.67 | 68.05 | 68.17 | 67.83
KNNH | 55.12 | 57.53 | 58.61 | 66.77 | 69.83 | 71.07
SCQ | 58.34 | 56.21 | 56.10 | 68.45 | 70.64 | 70.81
NRQ (Ours) | 60.83 | 62.08 | 63.34 | 69.91 | 72.07 | 73.82
SNRQ (Ours) | 61.78 | 62.74 | 62.60 | 70.83 | 72.33 | 73.37

Fig. 2: Training time for NRQ and SNRQ applied to the CIFAR-10 dataset for
different number of bits.

(a) Synthetic toy data

(b) ITQ

(c) SNRQ ($\alpha\\!=\\!3$. $\beta\\!=\\!50$)

(d) SNRQ ($\alpha\\!=\\!3$, $\beta\\!=\\!200$)

(e) SNRQ ($\alpha\\!=\\!2$, $\beta\\!=\\!50$)

(f) SNRQ ($\alpha\\!=\\!4$, $\beta\\!=\\!50$)

Fig. 3: Visualization of the ITQ & SNRQ performance in projecting the data to
a space with less quantization loss on a 2D toy dataset: (a) The reference 2-D
data, (b) data rotated & projected by ITQ, (c-f) Data rotated & projected by
SNRQ using different values of $\beta$ & $\alpha$. Data points are pushed
towards Hamming vertices corrupting the neighbour structure of data.

Comparison between NRQ and SNRQ: Here we compare NRQ and SNRQ both in terms of
run-time and performance. If we directly update the projection matrix, the
training becomes significantly time consuming. To address this, we proposed
sequential update scheme which achieves the same performance while
significantly reduces the time complexity. The training times of NRQ and SNRQ
for CIFAR-10 dataset are shown in Fig. 2. Although based on Fig. 2, SNRQ is
significantly faster compared with NRQ, as can be seen from Tables 1, 2, 3,
and 4 their performance is similar. The time difference for training may be
quite significant for petabyte archives with gigapixel files like satellite
imaging and digital pathology where images are commonly quite large, $>$
50,000 by 50,000 pixels. For the latter each patient often comes with many
images.

SNRQ vs. ITQ –  To validate how employing a non-rigid transformation acts
compared with rigid one, we generated a synthetic toy dataset. Fig. 3 shows
the transformed data by ITQ and SNRQ on this toy dataset under for values of
$\alpha$ and $\beta$. Fig. 3 part (a) represents the toy data. Fig. 3 part (b)
shows the same data after transformation by ITQ. Figs. 3 part (c), (d), (e),
and (f) show the transformed data using SNRQ for different values of $\alpha$
and $\beta$. This graph reveals role of these regularization parameters in
SNRQ algorithm which is controlling the trade-off between preserving
neighbourhood structure of data and pushing data points. They can cause unsafe
quantization. The smaller the $\beta$ (larger the $\alpha$) the unsafer the
quantization. The quantization loss is increasingly reduced by corrupting
neighbourhood.

Comparison of SNRQ with deep unsupervised hashing algorithms – Recently deep
learning has been applied to unsupervised hashing including DH Liong et al.
(2015), UHBDNN Do et al. (2016), DeepBit Lin et al. (2016), and SADH Shen et
al. (2018). To compare SNRQ with Deep unsupervised hashing methods, we employ
NUS-WIDE in the setting same as Table 4. As Table 5 shows, SNRQ outperforms
UHBDNN and DeepBit by a considerable gap. Compared with SADH, results are
highly competitive. Considering DeepBit and SADH are feeding images into a CNN
and learn the features, which implies that the quality of features used in
SADH, and DeepBit are expectedly better than VGG-F extracted features, it is
quite impressive that SNRQ is superior or delivering on par results.

Table 5: SNRQ compared with deep unsupervised hashing algorithms for NUS-WIDE dataset. The terms R+ and V+ mean the respective algorithm works on raw images and vector data (images after feature extraction) respectively.  | mAP % | precision % @5000
---|---|---
Method | 16 bits | 32 bits | 64 bits | 16 bits | 32 bits | 64 bits
V+UHBDN | 54.26 | 51.72 | 54.74 | 70.18 | 69.60 | 72.74
R+DeepBit | 39.22 | 40.32 | 42.06 | 45.54 | 51.34 | 57.72
R+SADH | 60.14 | 57.99 | 56.33 | 71.45 | 73.88 | 75.04
V+SNRQ | 61.78 | 62.74 | 62.60 | 70.83 | 72.33 | 73.37

Ablation study – We set $\alpha=3$ and $\beta=0.01$ for all experiments and
datasets. Based on the following experiments, although $\beta$ and $\alpha$
regulates the trade-off between preserving the neighbourhood structure and the
quality of quantization, SNRQ is robust to changes. Results in Table 6 show
the SNRQ performance on MNIST and LabelMe-12-50k datasets for different values
$\beta=0.01,0.05,0.1$, and $0.5$ for the 16 bit setting. Even the worst
performance after changing $\beta=0.01$ is still state-of-art. We also tested
the robustness of SNRQ to changes in $\alpha$. We should not use $\alpha=1$ as
it removes the $\mathbf{Q}$ from objective. For $\alpha$, first the $\beta$
value is set to 0.01, then we evaluated SNRQ for $\alpha=$2, 3, 4 and 5 in
Table 6. The performance for $\alpha=$2,3, 4 and 5 is consistently high
reaching highest for $\alpha=4$. Considering the proposed objective function
in Eq. 3, if we put $\alpha=0$, there will be no quantization step, and the
problem becomes a PCA-like objective function where the orthogonality
constraint has been smoothed. Similar to PCA, this would lead to poor binary
representations due to accumulated quantization error. For $\beta=0$, the
projection matrix $\mathbf{W}$ becomes too dominant and destroys the
neighbourhood of data leading to loss of information. Besides, from an
optimization point of view, for $\alpha=0$ and $\beta=0$, the objective
function would become a quadratic function where, given the fact that
covariance matrix is positive semi-definite, the maximization is unbounded. As
a result, $\alpha=0$ and $\beta=0$ do not make sense in the context of this
paper.

Achieving non-rigid projections – We achieved a non-rigid transformation by
relaxing orthogonality constraint on projection matrix. We used the well-known
soft orthogonality (SO)
$\|\mathbf{W}^{T}\mathbf{W}\\!-\\!\mathbf{I}_{K}\|^{2}_{F}$. However, there
are other ways of achieving this non-rigid projection matrix Bansal et al.
(2018). We also tried double soft orthogonality (DSO)
$\|\mathbf{W}^{T}\mathbf{W}\\!-\\!\mathbf{I}_{K}\|^{2}_{F}+\|\mathbf{W}\mathbf{W}^{T}\\!-\\!\mathbf{I}_{D}\|^{2}_{F}$,
and Mutual Coherence (MC)
$\|\mathbf{W}^{T}\mathbf{W}\\!-\\!\mathbf{I}_{K}\|^{2}_{\infty}$ where we
resorted to auto differentiation for implementation. Based on Table 7, the
simple soft orthogonality (SO) is achieving better performance in most cases.

Table 6: Comparison of retrieval performance for MNIST (GIST 512-D) and LabelMe-12-50k (4096-D VGG-FC7) datasets based on mAP for different values of $\beta$ and $\alpha$. |  16 bits, $\alpha=3$
---|---
Dataset | $\beta=0.01$ | $\beta=0.05$ | $\beta=0.1$ | $\beta=0.5$
MNIST | 64.70 | 64.29 | 67.17 | 66.38
LabelMe | 26.08 | 25.29 | 25.29 | 25.10
|  16 bits, $\beta=0.01$
Dataset | $\alpha=2$ | $\alpha=3$ | $\alpha=4$ | $\alpha=5$
MNIST | 63.3 | 63.18 | 65.58 | 65.30
LabelMe | 24.20 | 24.54 | 25.73 | 25.63
Table 7: Comparison of SO, DSO, and MC for obtaining non-rigid transformation on NCT-CRC-HE-100K dataset based on mAP and precision@1000. | mAP % | precision % @1000
---|---|---
Method | 16 bits | 32 bits | 64 bits | 16 bits | 32 bits | 64 bits
SO | 71.34 | 76.83 | 76.57 | 82.94 | 86.13 | 86.30
DSO | 71.07 | 76.29 | 75.40 | 81.59 | 85.60 | 86.16
MC | 72.43 | 73.94 | 76.86 | 82.70 | 83.72 | 85.86

##  4 Conclusions

We introduced Sequential Non-rigid Quantization (SNRQ). The backbone of SNRQ
is based on ITQ (iterative quantization) Gong et al. (2013). Although the
ideas presented in ITQ are interesting, we argued that learning projection and
rotation in two separate steps could be sub-optimal. Furthermore, a rigid
transformation may not be enough for reducing quantization to the ultimate
limit. Motivated by these limitations, we proposed an algorithm to reduce both
dimensionality and quantization loss simultaneously. We also employed a non-
rigid transformation to push for quantization beyond rotation. Employing non-
rigid transformations is generally against intuition. It does not preserve the
neighborhood of data (and all these efforts for reducing quantization error is
to preserve more neighborhood after binarization). However, we showed that
corrupting neighborhood in favor of reducing quantization eventually leads to
better codes. An efficient nested coordinate descent algorithm was employed to
update all three matrices. The results on five public datasets totaling almost
half a million images showed that the proposed method outperforms the state-
of-art linear hashing methods.

As the future work, we plan to extend the idea of binary representation
learning using the rigid and non-rigid transformations to the deep
architectures. In this framework, while the non-rigid transformations can be
realized using regularization terms similar to the soft orthogonality in Eq.
3. For the rigid transformation, one can project the gradient on to orthogonal
feasible set using the projection operation proposed in Manton (2002) to
update the rotation matrix. Besides, it has been recently demonstrated that
hashing can be used in the network quantization task Gholami et al. (2021).
This would be interesting to see how the proposed quantization method performs
for network quantization.

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#  $\Delta S=0$ Hadronic Parity Violation in Next-to-Leading Order QCD:
Anomalous Dimension Matrices
and Their Implications

Girish Muralidhara<EMAIL_ADDRESS>Susan Gardner
<EMAIL_ADDRESS>Department of Physics and Astronomy, University of
Kentucky, Lexington, Kentucky 40506-0055 USA

###### Abstract

We construct the effective Hamiltonian for hadronic parity violation in
strangeness-nonchanging ($\Delta S=0$) processes in next-to-leading order
(NLO) in QCD, for all isosectors, and at a renormalization scale of $2\,\rm
GeV$, thus extending our earlier leading-order (LO) analysis [1, 2]. Hadronic
parity violation, studied in the context of the low-energy interactions of
nucleons and nuclei, exposes the complex interplay of weak and strong
interactions in these systems, and thus supports our extension to NLO. Here we
exploit the flavor-blind nature of QCD interactions to construct the needed
anomalous dimension matrices from those computed in flavor physics, which we
then use to refine our effective Hamiltonian and finally our predicted parity-
violating meson-nucleon coupling constants, to find improved agreement with
few-body experiments.

##  1 Introduction

Hadronic parity violation (HPV) is associated with the low-energy, strangeness
(and heavy flavor)-nonchanging interactions of nucleons and nuclei, so that
its theoretical analysis within the Standard Model (SM) is a multi-scale
problem. The interpretation of the existing body of experimental results has
not yet crystallized into a complete and consistent picture, with the lack of
knowledge of the individual couplings that appear in chiral effective theory
descriptions [3, 4, 5, 6, 7, 8, 9, 10, 11], as well as nuclear structure
uncertainties [12, 13, 14, 15, 16], acting to muddy the picture. In recent
years, however, two important measurements of HPV have been completed: that of
the single-spin asymmetry in $\vec{n}+p\to d+\gamma$ [17] and in
$\vec{n}+{}^{3}{\rm He}\to p+{}^{3}{\rm H}$ [18], in few-nucleon systems for
which nuclear structure effects are under good control. In previous work we
have determined the effective Hamiltonian for HPV in the SM at a scale of 2
GeV using renormalization group (RG) methods in LO QCD [1]. Using this result
to match from a quark- to hadron-level description, in that we use it to
compute parity-violating meson-nucleon couplings for comparison to their
values extracted from the noted few-body experiments [17, 18], analyzed within
a one-meson-exchange model framework [19, 20], we find agreement within
$1\sigma$ of those experimental values [1, 2]. In this paper we thus focus on
refining the pertinent effective Hamiltonian for HPV in the SM at a
renormalization scale $\mu$ of 2 GeV. We do this, improving upon our earlier
LO analysis [1, 2], by working in NLO in QCD, evolving the complete effective
Hamiltonian — i.e., for all three isosectors — from the $W$ mass scale to the
2 GeV scale using RG techniques within perturbative QCD. Here we exploit the
extensive studies of NLO corrections in flavor-changing non-leptonic decays in
[21, 22] and reviewed in [23] to bring this to pass. With this improvement in
place, we update our earlier parity-violating meson-nucleon coupling constant
assessments to find improved agreement with the experimental determinations we
have noted, thus arguing for the use of our effective Hamiltonian in a
systematic assessment of the low-energy couplings in chiral effective theory
in future work.

Previous work along these lines has focused on the isovector ($I=1$) sector,
as the role of parity-violating $\pi^{\pm}$ exchange in the nucleon-nucleon
force was once thought to be dominant [19]. However, various lines of evidence
are at odds with that picture, including the non-observation of parity
violation in 18F radiative decay [12, 13, 14], the outcomes of a large-$N_{c}$
analysis of pionless chiral effective theory [24, 25, 26], as well as the
results of the recent few-body measurements themselves [17, 18], thus
suggesting that isoscalar, isovector, and isotensor sectors can all play
significant phenomenological roles — and thus all must be considered. For
context, we note earlier determinations of the anomalous dimension matrix in
our current context in LO QCD in the isovector sector [27, 28], as well as
work in NLO QCD in the isovector [29] and isotensor [30] sectors. In this
paper we analyze the odd ($I=1$) and even ($I=0\oplus 2$) isosectors
separately, noting that an operator of pure $I=2$ can only be formed if the
Cabibbo angle is set to zero, as we discuss in Sec. 3.

We now explain how we are able to adapt the earlier work in flavor physics to
our current purpose. In [21], the two-loop current-current corrections are
computed in different regularization schemes, and it is shown that the final
effective Hamiltonian is independent of the scheme even though the
intermediate stages of calculations are not. In [22], the anomalous
dimensional matrices of the $|\Delta F|=1\,(F=S,C,B)$ four-quark operators are
calculated by inserting them into two-loop current-current as well as gluonic
penguins (both types) and electroweak penguin diagrams. Since the QCD
corrections are flavor blind up to quark mass effects, a general mixing scheme
for chiral four-quark operators should exist. At LO, [31] gives a dictionary
for the mixing in different categories of four-quark operators. This
dictionary is independent of the nature of processes, and anomalous matrices
can be extracted for either flavor conserving or non-conserving systems.
Conversely, given adequate information on mixing in one particular theory, the
mixing scheme for prototype four-quark operators can be reconstructed and
utilized for obtaining the anomalous dimensions in a different theory. In line
with this, the first step is to extract the general mixing scheme of flavor-
conserving, four-quark structures from flavor non-conserving studies. This was
attempted by [29] using NLO studies in flavor physics in the ’t Hooft-Veltman
scheme [22]. Although we agree with the proposed framework, the provided set
of operators is not complete and, moreover, crucial features, such as the
partial cancellation of certain penguin contributions, are missing. We also
follow this general path, but our method also differs in that we work in an
explicit parity violating basis ($V\otimes A$-like structures) instead of
working in a chiral basis ($(V-A)\otimes(V+A)$-like structures). This enables
us to determine the aforementioned cancellation effects more easily.

The structure of the paper is as follows. In Sec. 2 we discuss how we
determine the anomalous dimension matrix and its associated complete set of
operators, and we develop the inclusion of current-current and penguin
corrections in detail. In Sec. 3, we present the form of the anomalous
dimension matrices in the different isosectors. Then, in Sec. 4 we summarize
NLO RG flow theory, evaluating the Wilson coefficients (WCs) for the
individual isosectors and using them to compute the parity-violating meson-
nucleon couplings, following the methods and inputs of [1, 2]. Finally, we
conclude in Sec. 5 by offering a summary and outlook.

##  2 Anomalous dimension matrices

We denote the anomalous dimension matrices for in LO and NLO as

$\gamma(\alpha_{s})=\gamma_{\rm LO}+\gamma_{\rm
NLO}\equiv\frac{\alpha_{s}}{4\pi}\gamma^{(0)}+\left(\frac{\alpha_{s}}{4\pi}\right)^{2}\gamma^{(1)}\,.$
(1)

We focus on QCD corrections, noting that a consistent treatment of electroweak
corrections implies an expansion of the operator basis [22] with outcomes that
are of smaller numerical size (in $|\Delta F|=1$ decays) than the effects we
consider [23]. In what follows, we drop the subscripts on $\gamma$-matrices,
with “NLO” to be understood unless noted otherwise. We introduce prototype
operators [29, 31] that facilitate the connection between flavor-changing and
flavor-conserving sectors, use these to determine the anomalous dimensions for
current-current or penguin operator insertions, as shown in Fig. 1, and then
adapt those results for use in HPV.

$\psi_{2}$$\psi_{1}$$q$$q$ (a) Type-1

$\psi_{2}$$\psi_{1}$$q$$q$ (b) Type-2

Figure 1:  (a) “Type-1” penguins with possible
$\bar{\psi}_{1}\psi_{2}(\bar{\psi}_{3}\psi_{3})$ fermion operator insertions
versus (b) “type-2” penguins with possible
$\bar{\psi}_{1}(\psi_{3}\bar{\psi}_{3})\psi_{2}$ insertions, as shown here in
LO QCD.

###  2.1 Prototype Current-Current Corrections

In a chiral basis, the prototype operator set $\vec{Q}$ for mixing via
current-current ($cc$) insertions is [31, 29]

$\begin{smallmatrix}Q_{1}=(\bar{\psi}_{1}\psi_{2})_{V-A}^{\alpha\alpha}(\bar{\psi}_{3}\psi_{4})_{V-A}^{\beta\beta}-(\bar{\psi}_{1}\psi_{2})_{V+A}^{\alpha\alpha}(\bar{\psi}_{3}\psi_{4})_{V+A}^{\beta\beta}\,\,&Q_{2}=(\bar{\psi}_{1}\psi_{2})_{V-A}^{\alpha\beta}(\bar{\psi}_{3}\psi_{4})_{V-A}^{\beta\alpha}-(\bar{\psi}_{1}\psi_{2})_{V+A}^{\alpha\beta}(\bar{\psi}_{3}\psi_{4})_{V+A}^{\beta\alpha}\\\
Q_{3}=(\bar{\psi}_{1}\psi_{2})_{V-A}^{\alpha\alpha}(\bar{\psi}_{3}\psi_{4})_{V+A}^{\beta\beta}-(\bar{\psi}_{1}\psi_{2})_{V+A}^{\alpha\alpha}(\bar{\psi}_{3}\psi_{4})_{V-A}^{\beta\beta}\,\,&Q_{4}=(\bar{\psi}_{1}\psi_{2})_{V-A}^{\alpha\beta}(\bar{\psi}_{3}\psi_{4})_{V+A}^{\beta\alpha}-(\bar{\psi}_{1}\psi_{2})_{V+A}^{\alpha\beta}(\bar{\psi}_{3}\psi_{4})_{V-A}^{\beta\alpha}\end{smallmatrix}\,,$
(2)

where the superscripts are SU(3) color indices with
$(\bar{\psi}_{1}\psi_{2})^{\alpha\beta}_{V\mp
A}=\bar{\psi}_{1}^{\alpha}\gamma_{\mu}(1\mp\gamma_{5})\psi_{2}^{\beta}\equiv
V_{\stackrel{{\scriptstyle L}}{{{}_{R}}}}$, which mix at NLO via the matrix
[22]:

$\gamma_{\vec{Q}}^{cc}=\left(\frac{\alpha_{s}}{4\pi}\right)^{2}\left(\begin{smallmatrix}\frac{553}{6}-\frac{58f}{9}&\frac{95}{2}-2f&0&0\\\
\frac{95}{2}-2f&\frac{553}{6}-\frac{58f}{9}&0&0\\\
0&0&121-\frac{62f}{9}&-\frac{2f}{3}-39\\\
0&0&\frac{95}{2}-\frac{4f}{3}&-\frac{44f}{9}-\frac{85}{2}\\\
\end{smallmatrix}\right)\,,$ (3)

where the anomalous dimension of the second set of operators in each $Q_{i}$
follow from the first [22] under the parity symmetry of the QCD corrections.
Here $f$ denotes the number of dynamical quarks in the theory; it thus varies
as we run down the energy scale. We can convert the chiral basis to one of
$V\otimes A$ or $A\otimes V$ form after a 45∘ rotation111The matrix
$R_{\theta}$ is a nested combination of rotation matrices
$\bigl{(}\begin{smallmatrix}\cos\theta&-\sin\theta\\\
\sin\theta&\cos\theta\end{smallmatrix}\bigr{)}$ in color-singlet and
nonsinglet operators.: $\vec{\Phi}\propto R_{\frac{\pi}{4}}\vec{Q}$, to yield

$\begin{smallmatrix}\Phi_{1}=(\bar{\psi}_{1}\psi_{2})_{V}^{\alpha\alpha}(\bar{\psi}_{3}\psi_{4})_{A}^{\beta\beta}\,\,\,\,\,\,\,&\Phi_{2}=(\bar{\psi}_{1}\psi_{2})_{V}^{\alpha\beta}(\bar{\psi}_{3}\psi_{4})_{A}^{\beta\alpha}\\\
\Phi_{3}=(\bar{\psi}_{1}\psi_{2})_{A}^{\alpha\alpha}(\bar{\psi}_{3}\psi_{4})_{V}^{\beta\beta}\,\,\,\,\,\,\,&\Phi_{4}=(\bar{\psi}_{1}\psi_{2})_{A}^{\alpha\beta}(\bar{\psi}_{3}\psi_{4})_{V}^{\beta\alpha}\end{smallmatrix}$
(4)

with the corresponding anomalous dimension matrix obtained by
$R_{\frac{\pi}{4}}\gamma_{\vec{Q}}^{cc}R^{T}_{\frac{\pi}{4}}$:

$\mathcal{C}=\left(\frac{\alpha_{s}}{4\pi}\right)^{2}\left(\begin{smallmatrix}\frac{1279}{12}-\frac{20f}{3}&\frac{17}{4}-\frac{4f}{3}&\frac{2f}{9}-\frac{173}{12}&\frac{173}{4}-\frac{2f}{3}\\\
\frac{95}{2}-\frac{5f}{3}&\frac{149}{6}-\frac{17f}{3}&-\frac{f}{3}&\frac{202}{3}-\frac{7f}{9}\\\
\frac{2f}{9}-\frac{173}{12}&\frac{173}{4}-\frac{2f}{3}&\frac{1279}{12}-\frac{20f}{3}&\frac{17}{4}-\frac{4f}{3}\\\
-\frac{f}{3}&\frac{202}{3}-\frac{7f}{9}&\frac{95}{2}-\frac{5f}{3}&\frac{149}{6}-\frac{17f}{3}\\\
\end{smallmatrix}\right)\,.$ (5)

###  2.2 Prototype Penguin Corrections

Inserting operators
$(\bar{\psi}_{1}\psi_{2})_{V-A}^{\alpha\alpha}(\bar{\psi}_{3}\psi_{3})_{V-A}^{\beta\beta}$
and
$(\bar{\psi}_{1}\psi_{2})_{V-A}^{\alpha\beta}(\bar{\psi}_{3}\psi_{3})_{V-A}^{\beta\alpha}$
into type-1 ($p1$) penguins and
$(\bar{\psi}_{1}\psi_{3})_{V-A}^{\alpha\alpha}(\bar{\psi}_{3}\psi_{2})_{V-A}^{\beta\beta}$
and
$(\bar{\psi}_{1}\psi_{3})_{V-A}^{\alpha\beta}(\bar{\psi}_{3}\psi_{2})_{V-A}^{\beta\alpha}$
into type-2 ($p2$) penguins generate operators of $V_{L}\otimes V_{L}$ and
$V_{L}\otimes V_{R}$ form [22], so as in the $cc$ case, we form the prototype
operator set $\vec{P}$

$\begin{smallmatrix}P_{1}=(\bar{\psi}_{1}\psi_{2})_{V-A}^{\alpha\alpha}\sum_{q}^{f}(\bar{q}q)_{V+A}^{\beta\beta}-(\bar{\psi}_{1}\psi_{2})_{V+A}^{\alpha\alpha}\sum_{q}^{f}(\bar{q}q)_{V-A}^{\beta\beta}\,\,&P_{2}=(\bar{\psi}_{1}\psi_{2})_{V-A}^{\alpha\beta}\sum_{q}^{f}(\bar{q}q)_{V+A}^{\beta\alpha}-(\bar{\psi}_{1}\psi_{2})_{V+A}^{\alpha\beta}\sum_{q}^{f}(\bar{q}q)_{V-A}^{\beta\alpha}\\\
P_{3}=(\bar{\psi}_{1}\psi_{2})_{V-A}^{\alpha\alpha}\sum_{q}^{f}(\bar{q}q)_{V-A}^{\beta\beta}-(\bar{\psi}_{1}\psi_{2})_{V+A}^{\alpha\alpha}\sum_{q}^{f}(\bar{q}q)_{V+A}^{\beta\beta}\,\,&P_{4}=(\bar{\psi}_{1}\psi_{2})_{V-A}^{\alpha\beta}\sum_{q}^{f}(\bar{q}q)_{V-A}^{\beta\alpha}-(\bar{\psi}_{1}\psi_{2})_{V+A}^{\alpha\beta}\sum_{q}^{f}(\bar{q}q)_{V+A}^{\beta\alpha}\end{smallmatrix}$
(6)

with the parity symmetry of the QCD corrections guaranteeing that each term of
$P_{i}$ have the same mixing profile. The anomalous matrices due to insertions
in type-1 ($p1$) and type-2 ($p2$) penguins, determined from [22], are

$\gamma_{\vec{P}}^{p1}=f\left(\frac{\alpha_{s}}{4\pi}\right)^{2}\left(\begin{smallmatrix}\frac{71}{9}&\frac{1}{3}&-\frac{73}{9}&\frac{1}{3}\\\
-\frac{256}{243}&\frac{832}{81}&-\frac{1246}{243}&\frac{382}{81}\\\
-\frac{25}{3}&1&\frac{23}{3}&1\\\
-\frac{1210}{243}&\frac{490}{81}&-\frac{418}{243}&\frac{850}{81}\\\
\end{smallmatrix}\right)\,;$

${\gamma_{\vec{P}}}^{p2}=2\left(\frac{\alpha_{s}}{4\pi}\right)^{2}\left(\begin{smallmatrix}0&0&0&0\\\
0&0&0&0\\\ -\frac{1210}{243}&\frac{490}{81}&-\frac{418}{243}&\frac{850}{81}\\\
-\frac{25}{3}&1&\frac{23}{3}&1\\\ \end{smallmatrix}\right)\,.$ (7)

Transforming to a basis of $V\otimes A$ or $A\otimes V$ form through
$\vec{\Phi}\propto R_{-\frac{\pi}{4}}\vec{P}$ we have:

$\begin{smallmatrix}\Phi_{1}=(\bar{\psi}_{1}\psi_{2})_{A}^{\alpha\alpha}\sum_{q}^{f}(\bar{q}q)_{V}^{\beta\beta}\,\,\,\,&\Phi_{2}=(\bar{\psi}_{1}\psi_{2})_{A}^{\alpha\beta}\sum_{q}^{f}(\bar{q}q)_{V}^{\beta\alpha}\\\
\Phi_{3}=(\bar{\psi}_{1}\psi_{2})_{V}^{\alpha\alpha}\sum_{q}^{f}(\bar{q}q)_{A}^{\beta\beta}\,\,\,\,&\Phi_{4}=(\bar{\psi}_{1}\psi_{2})_{V}^{\alpha\beta}\sum_{q}^{f}(\bar{q}q)_{A}^{\beta\alpha}\end{smallmatrix}$
(8)

with anomalous dimension matrices determined by
$R_{-\frac{\pi}{4}}\gamma_{\vec{P}}^{p1,p2}R^{T}_{-\frac{\pi}{4}}$:

$\stackrel{{\scriptstyle\sim}}{{\smash{\mathcal{P}_{1}}\rule{0.0pt}{4.73611pt}}}=f\left(\frac{\alpha_{s}}{4\pi}\right)^{2}\left(\begin{smallmatrix}-\frac{4}{9}&\frac{4}{3}&0&0\\\
-\frac{1565}{243}&\frac{1277}{81}&-\frac{11}{27}&-\frac{5}{9}\\\
-\frac{2}{9}&\frac{2}{3}&16&0\\\ -\frac{7}{27}&\frac{7}{9}&\frac{11}{3}&5\\\
\end{smallmatrix}\right)\,;$

$\stackrel{{\scriptstyle\sim}}{{\smash{\mathcal{P}_{2}}\rule{0.0pt}{4.73611pt}}}=2\left(\frac{\alpha_{s}}{4\pi}\right)^{2}\left(\begin{smallmatrix}-\frac{814}{243}&\frac{670}{81}&\frac{44}{27}&\frac{20}{9}\\\
-\frac{1}{3}&1&8&0\\\
-\frac{814}{243}&\frac{670}{81}&\frac{44}{27}&\frac{20}{9}\\\
-\frac{1}{3}&1&8&0\\\ \end{smallmatrix}\right)\,,$ (9)

where
${\mathcal{P}}_{1}\equiv\,\stackrel{{\scriptstyle\sim}}{{\smash{{\mathcal{P}}_{1}}\rule{0.0pt}{4.73611pt}}}/f$,
${\mathcal{P}}_{2}\equiv\,\stackrel{{\scriptstyle\sim}}{{\smash{{\mathcal{P}}_{2}}\rule{0.0pt}{4.73611pt}}}/2$
for future use.

###  2.3 HPV Current-Current Corrections

The complete set of HPV operators from $Z^{0}$ exchange in terms of u, d, and
s quarks is [1]

$\footnotesize\begin{split}\Theta_{1}&=[(\bar{u}u)_{V}+(\bar{d}d)_{V}+(\bar{s}s)_{V}]^{\alpha\alpha}[(\bar{u}u)_{A}-(\bar{d}d)_{A}-(\bar{s}s)_{A}]^{\beta\beta}\\\
\Theta_{2}&=[(\bar{u}u)_{V}+(\bar{d}d)_{V}+(\bar{s}s)_{V}]^{\alpha\beta}[(\bar{u}u)_{A}-(\bar{d}d)_{A}-(\bar{s}s)_{A}]^{\beta\alpha}\\\
\Theta_{3}&=[(\bar{u}u)_{A}+(\bar{d}d)_{A}+(\bar{s}s)_{A}]^{\alpha\alpha}[(\bar{u}u)_{V}-(\bar{d}d)_{V}-(\bar{s}s)_{V}]^{\beta\beta}\\\
\Theta_{4}&=[(\bar{u}u)_{A}+(\bar{d}d)_{A}+(\bar{s}s)_{A}]^{\alpha\beta}[(\bar{u}u)_{V}-(\bar{d}d)_{V}-(\bar{s}s)_{V}]^{\beta\alpha}\\\
\Theta_{5}&=[(\bar{u}u)_{V}-(\bar{d}d)_{V}-(\bar{s}s)_{V}]^{\alpha\alpha}[(\bar{u}u)_{A}-(\bar{d}d)_{A}-(\bar{s}s)_{A}]^{\beta\beta}\\\
\Theta_{6}&=[(\bar{u}u)_{V}-(\bar{d}d)_{V}-(\bar{s}s)_{V}]^{\alpha\beta}[(\bar{u}u)_{A}-(\bar{d}d)_{A}-(\bar{s}s)_{A}]^{\beta\alpha}\\\
\Theta_{7}&=[(\bar{u}u)_{A}+(\bar{d}d)_{A}+(\bar{s}s)_{A}]^{\alpha\alpha}[(\bar{u}u)_{V}+(\bar{d}d)_{V}+(\bar{s}s)_{V}]^{\beta\beta}\\\
\Theta_{8}&=[(\bar{u}u)_{A}+(\bar{d}d)_{A}+(\bar{s}s)_{A}]^{\alpha\beta}[(\bar{u}u)_{V}+(\bar{d}d)_{V}+(\bar{s}s)_{V}]^{\beta\alpha}\,.\end{split}$
(10)

We can extend this set to include heavier quarks by respecting the structure
shared by u-like and d-like quarks [27, 1]. The first four operators come from
a prototype chiral basis color singlet and color non-singlet, $V_{L}\otimes
V_{L}$ and $V_{L}\otimes V_{R}$ structures, while the next four operators
follow from a prototype chiral basis with color singlet and color non-singlet,
$V_{L}\otimes V_{L}$ structures. The $cc$ anomalous dimension matrix for Eq.
(10) follows from Eq. (5):

$\gamma^{HPV}_{cc}=\left(\frac{\alpha_{s}}{4\pi}\right)^{2}\left(\begin{smallmatrix}\frac{1279}{12}-\frac{20f}{3}&\frac{17}{4}-\frac{4f}{3}&\frac{2f}{9}-\frac{173}{12}&\frac{173}{4}-\frac{2f}{3}&0&0&0&0\\\
\frac{95}{2}-\frac{5f}{3}&\frac{149}{6}-\frac{17f}{3}&-\frac{f}{3}&\frac{202}{3}-\frac{7f}{9}&0&0&0&0\\\
\frac{2f}{9}-\frac{173}{12}&\frac{173}{4}-\frac{2f}{3}&\frac{1279}{12}-\frac{20f}{3}&\frac{17}{4}-\frac{4f}{3}&0&0&0&0\\\
-\frac{f}{3}&\frac{202}{3}-\frac{7f}{9}&\frac{95}{2}-\frac{5f}{3}&\frac{149}{6}-\frac{17f}{3}&0&0&0&0\\\
0&0&0&0&\frac{553}{6}-\frac{58f}{9}&\frac{95}{2}-2f&0&0\\\
0&0&0&0&\frac{95}{2}-2f&\frac{553}{6}-\frac{58f}{9}&0&0\\\
0&0&0&0&0&0&\frac{553}{6}-\frac{58f}{9}&\frac{95}{2}-2f\\\
0&0&0&0&0&0&\frac{95}{2}-2f&\frac{553}{6}-\frac{58f}{9}\\\
\end{smallmatrix}\right)\,.$ (11)

###  2.4 HPV Penguin Corrections

The penguin corrections follow by inserting operators of
$\bar{\psi}_{1}\psi_{2}(\bar{\psi}_{3}\psi_{3})$ form in a type-1 penguin and
those of $\bar{\psi}_{1}(\psi_{3}\bar{\psi}_{3})\psi_{2}$ form in a type-2
penguin, as in Fig. 1. Thus for each operator in Eq. (10), a term of form
$(\bar{q}q)_{V}(\bar{q}q)_{A}$ can be inserted in either a type-1 or type-2
penguin to yield operators of form
$({\bar{q}}q)_{V}\sum_{f}({\bar{q}_{f}}q_{f})_{A}$ or
$({\bar{q}}q)_{A}\sum_{f}({\bar{q}_{f}}q_{f})_{V}$ of either color structure.
Also a term of form $(\bar{q}q)_{V}(\bar{q}^{\prime}q^{\prime})_{A}$ can be
inserted in a type-1 penguin to yield
$({\bar{q}}q)_{V}\sum_{f}({\bar{q}_{f}}q_{f})_{A}$ or
$(\bar{q}^{\prime}q^{\prime})_{A}\sum_{f}({\bar{q}_{f}}q_{f})_{V}$ of either
color structure. Moreover, odd (1,3) and even (2,4) rows of the matrix
$\mathcal{P}_{2}$ are identical. This effectively makes the type-2 penguin
contributions two fold. Thus we have

$\footnotesize\begin{split}[\gamma(\Theta_{1})]_{1j}\xrightarrow[corrections]{penguin}&\left(\sum_{i=1}^{4}\left(f\cdot\left(\mathcal{P}_{1}\right)_{1i}+2\cdot\left(\mathcal{P}_{2}\right)_{1i}\right)\Theta_{i}\right)-q\cdot\left(\left(\mathcal{P}_{1}\right)_{31}+\left(\mathcal{P}_{1}\right)_{33}\right)\Theta_{7}-q\cdot\left(\left(\mathcal{P}_{1}\right)_{32}+\left(\mathcal{P}_{1}\right)_{34}\right)\Theta_{8}\end{split}$
(12)

where $q=n_{d}-n_{u}$ is the difference in the number of open $d$-like and
$u$-like flavors. This originates due to the relative sign difference between
the $u$-like and $d$-like structures in Eq. (10), yielding cancellations, that
have also been shown through direct calculation in LO [1]. Similar results
hold for $\Theta_{2},\,\Theta_{3}$, and $\Theta_{4}$. Next we consider the
operators from the second block of operators:

$\footnotesize\begin{split}[\gamma(\Theta_{5})]_{5j}\xrightarrow[corrections]{penguin}&\left(\sum_{i=1}^{4}(-q)\cdot\left(\left(\mathcal{P}_{1}\right)_{1i}+\left(\mathcal{P}_{1}\right)_{3i}\right)\Theta_{i}\right)+2\cdot\left(\left(\mathcal{P}_{2}\right)_{11}+\left(\mathcal{P}_{2}\right)_{13}\right)\Theta_{7}+2\cdot\left(\left(\mathcal{P}_{2}\right)_{12}+\left(\mathcal{P}_{2}\right)_{14}\right)\Theta_{8}\end{split}$
(13)

with similar mixing for $\Theta_{6}$. The last two are highly symmetric
operators with

$\footnotesize\begin{split}[\gamma(\Theta_{7})]_{7j}\xrightarrow[corrections]{penguin}&\left[f\cdot\left((\mathcal{P}_{1})_{11}+(\mathcal{P}_{1})_{31}+(\mathcal{P}_{1})_{13}+(\mathcal{P}_{1})_{33}\right)+2\cdot\left((\mathcal{P}_{2})_{11}+(\mathcal{P}_{2})_{13}\right)\right]\Theta_{7}\\\
&+\left[f\cdot\left((\mathcal{P}_{1})_{12}+(\mathcal{P}_{1})_{32}+(\mathcal{P}_{1})_{14}+(\mathcal{P}_{1})_{34}\right)+2\cdot\left((\mathcal{P}_{2})_{12}+(\mathcal{P}_{2})_{14}\right)\right]\Theta_{8}\end{split}$
(14)

and similar mixing for $\Theta_{8}$. Consolidating this for the full basis we
have:

$\gamma^{HPV}_{penguin}=\left(\frac{\alpha_{s}}{4\pi}\right)^{2}\left(\begin{smallmatrix}-\frac{4f}{9}-\frac{1628}{243}&\frac{4f}{3}+\frac{1340}{81}&\frac{88}{27}&\frac{40}{9}&0&0&-\frac{142q}{9}&-\frac{2q}{3}\\\
-\frac{1565f}{243}-\frac{2}{3}&\frac{1277f}{81}+2&16-\frac{11f}{27}&-\frac{5f}{9}&0&0&-\frac{92q}{27}&-\frac{52q}{9}\\\
-\frac{2f}{9}-\frac{1628}{243}&\frac{2f}{3}+\frac{1340}{81}&16f+\frac{88}{27}&\frac{40}{9}&0&0&\frac{4q}{9}&-\frac{4q}{3}\\\
-\frac{7f}{27}-\frac{2}{3}&\frac{7f}{9}+2&\frac{11f}{3}+16&5f&0&0&\frac{1664q}{243}&-\frac{1232q}{81}\\\
\frac{2q}{3}&-2q&-16q&0&0&0&-\frac{836}{243}&\frac{1700}{81}\\\
\frac{1628q}{243}&-\frac{1340q}{81}&-\frac{88q}{27}&-\frac{40q}{9}&0&0&\frac{46}{3}&2\\\
0&0&0&0&0&0&\frac{46f}{3}-\frac{836}{243}&2f+\frac{1700}{81}\\\
0&0&0&0&0&0&\frac{46}{3}-\frac{836f}{243}&\frac{1700f}{81}+2\\\
\end{smallmatrix}\right)\,.$ (15)

Finally, we combine the current and penguin contributions to get the complete
matrix, $\gamma_{NLO}^{HPV}=\gamma^{HPV}_{cc}+\gamma^{HPV}_{penguin}$, namely,

$\gamma_{NLO}^{HPV}=\left(\frac{\alpha_{s}}{4\pi}\right)^{2}\left(\begin{smallmatrix}\frac{97087}{972}-\frac{64f}{9}&\frac{6737}{324}&\frac{2f}{9}-\frac{1205}{108}&\frac{1717}{36}-\frac{2f}{3}&0&0&-\frac{142q}{9}&-\frac{2q}{3}\\\
\frac{281}{6}-\frac{1970f}{243}&\frac{818f}{81}+\frac{161}{6}&16-\frac{20f}{27}&\frac{202}{3}-\frac{4f}{3}&0&0&-\frac{92q}{27}&-\frac{52q}{9}\\\
-\frac{20525}{972}&\frac{19373}{324}&\frac{28f}{3}+\frac{11863}{108}&\frac{313}{36}-\frac{4f}{3}&0&0&\frac{4q}{9}&-\frac{4q}{3}\\\
-\frac{16f+18}{27}&\frac{208}{3}&2f+\frac{127}{2}&\frac{149-4f}{6}&0&0&\frac{1664q}{243}&-\frac{1232q}{81}\\\
\frac{2q}{3}&-2q&-16q&0&\frac{553}{6}-\frac{58f}{9}&\frac{95}{2}-2f&-\frac{836}{243}&\frac{1700}{81}\\\
\frac{1628q}{243}&-\frac{1340q}{81}&-\frac{88q}{27}&-\frac{40q}{9}&\frac{95}{2}-2f&\frac{553}{6}-\frac{58f}{9}&\frac{46}{3}&2\\\
0&0&0&0&0&0&\frac{80f}{9}+\frac{43121}{486}&\frac{11095}{162}\\\
0&0&0&0&0&0&\frac{377}{6}-\frac{1322f}{243}&\frac{1178f}{81}+\frac{565}{6}\\\
\end{smallmatrix}\right)\,.$ (16)

##  3 Isosector extractions

We now study $\gamma_{NLO}^{HPV}$ in various isosectors. The $Z^{0}$-sector
operators of isovector HPV are [27, 1]:

$\footnotesize\begin{split}\Theta_{1}&{}^{I=1}=[(\bar{u}u)_{V}+(\bar{d}d)_{V}+(\bar{s}s)_{V}]^{\alpha\alpha}[(\bar{u}u)_{A}-(\bar{d}d)_{A}]^{\beta\beta}\\\
\Theta_{2}&{}^{I=1}=[(\bar{u}u)_{V}+(\bar{d}d)_{V}+(\bar{s}s)_{V}]^{\alpha\beta}[(\bar{u}u)_{A}-(\bar{d}d)_{A}]^{\beta\alpha}\\\
\Theta_{3}&{}^{I=1}=[(\bar{u}u)_{A}+(\bar{d}d)_{A}+(\bar{s}s)_{A}]^{\alpha\alpha}[(\bar{u}u)_{V}-(\bar{d}d)_{V}]^{\beta\beta}\\\
\Theta_{4}&{}^{I=1}=[(\bar{u}u)_{A}+(\bar{d}d)_{A}+(\bar{s}s)_{A}]^{\alpha\beta}[(\bar{u}u)_{V}-(\bar{d}d)_{V}]^{\beta\alpha}\\\
\Theta_{5}&{}^{I=1}=(\bar{s}s)_{V}^{\alpha\alpha}[(\bar{u}u)_{A}-(\bar{d}d)_{A}]^{\beta\beta}\\\
\Theta_{6}&{}^{I=1}=(\bar{s}s)_{V}^{\alpha\beta}[(\bar{u}u)_{A}-(\bar{d}d)_{A}]^{\beta\alpha}\\\
\Theta_{7}&{}^{I=1}=(\bar{s}s)_{A}^{\alpha\alpha}[(\bar{u}u)_{V}-(\bar{d}d)_{V}]^{\beta\beta}\\\
\Theta_{8}&{}^{I=1}=(\bar{s}s)_{A}^{\alpha\beta}[(\bar{u}u)_{V}-(\bar{d}d)_{V}]^{\beta\alpha}\\\
\end{split}\,.$ (17)

Following the procedures of the previous section, the isovector anomalous
dimension matrix becomes:

$\gamma_{NLO}^{I=1}=\left(\frac{\alpha_{s}}{4\pi}\right)^{2}\left(\begin{smallmatrix}\frac{97087}{972}-\frac{64f}{9}&\frac{6737}{324}&\frac{2f}{9}-\frac{1205}{108}&\frac{1717}{36}-\frac{2f}{3}&0&0&0&0\\\
\frac{281}{6}-\frac{1970f}{243}&\frac{818f}{81}+\frac{161}{6}&16-\frac{20f}{27}&\frac{202}{3}-\frac{4f}{3}&0&0&0&0\\\
-\frac{20525}{972}&\frac{19373}{324}&\frac{28f}{3}+\frac{11863}{108}&\frac{313}{36}-\frac{4f}{3}&0&0&0&0\\\
\frac{-16f-18}{27}&\frac{208}{3}&2f+\frac{127}{2}&\frac{149-4f}{6}&0&0&0&0\\\
-\frac{4q}{9}&\frac{4q}{3}&0&0&\frac{1279-80f}{12}&\frac{17}{4}-\frac{4f}{3}&\frac{8f-519}{36}&\frac{173}{4}-\frac{2f}{3}\\\
-\frac{1565q}{243}&\frac{1277q}{81}&-\frac{11q}{27}&-\frac{5q}{9}&\frac{95}{2}-\frac{5f}{3}&\frac{149-34f}{6}&-\frac{f}{3}&\frac{606-7f}{9}\\\
-\frac{2q}{9}&\frac{2q}{3}&16q&0&\frac{8f-519}{36}&\frac{173}{4}-\frac{2f}{3}&\frac{1279-80f}{12}&\frac{17}{4}-\frac{4f}{3}\\\
-\frac{7q}{27}&\frac{7q}{9}&\frac{11q}{3}&5q&-\frac{f}{3}&\frac{606-7f}{9}&\frac{95}{2}-\frac{5f}{3}&\frac{149-34f}{6}\\\
\end{smallmatrix}\right)\,.$ (18)

In contrast, the operators of $I=0\oplus 2$ sector are [1]:

$\footnotesize\begin{split}\Theta_{1}&{}^{I=0\oplus
2}=[(\bar{u}u)_{V}+(\bar{d}d)_{V}+(\bar{s}s)_{V}]^{\alpha\alpha}[(\bar{s}s)_{A}]^{\beta\beta}\\\
\Theta_{2}&{}^{I=0\oplus
2}=[(\bar{u}u)_{V}+(\bar{d}d)_{V}+(\bar{s}s)_{V}]^{\alpha\beta}[(\bar{s}s)_{A}]^{\beta\alpha}\\\
\Theta_{3}&{}^{I=0\oplus
2}=[(\bar{u}u)_{A}+(\bar{d}d)_{A}+(\bar{s}s)_{A}]^{\alpha\alpha}[(\bar{s}s)_{V}]^{\beta\beta}\\\
\Theta_{4}&{}^{I=0\oplus
2}=[(\bar{u}u)_{A}+(\bar{d}d)_{A}+(\bar{s}s)_{A}]^{\alpha\beta}[(\bar{s}s)_{V}]^{\beta\alpha}\\\
\Theta_{5}&{}^{I=0\oplus
2}=[(\bar{u}u)_{V}-(\bar{d}d)_{V}]^{\alpha\alpha}[(\bar{u}u)_{A}-(\bar{d}d)_{A}]^{\beta\beta}+(\bar{s}s)_{V}^{\alpha\alpha}(\bar{s}s)_{A}^{\beta\beta}\\\
\Theta_{6}&{}^{I=0\oplus
2}=[(\bar{u}u)_{V}-(\bar{d}d)_{V}]^{\alpha\beta}[(\bar{u}u)_{A}-(\bar{d}d)_{A}]^{\beta\alpha}+(\bar{s}s)_{V}^{\alpha\beta}(\bar{s}s)_{A}^{\beta\alpha}\\\
\Theta_{7}&{}^{I=0\oplus
2}=[(\bar{u}u)_{V}+(\bar{d}d)_{V}+(\bar{s}s)_{V}]^{\alpha\alpha}[(\bar{u}u)_{A}+(\bar{d}d)_{A}+(\bar{s}s)_{A}]^{\beta\beta}\\\
\Theta_{8}&{}^{I=0\oplus
2}=[(\bar{u}u)_{A}+(\bar{d}d)_{A}+(\bar{s}s)_{A}]^{\alpha\beta}[(\bar{u}u)_{V}+(\bar{d}d)_{V}+(\bar{s}s)_{V}]^{\beta\alpha}\\\
\end{split}\,,$ (19)

and the corresponding anomalous matrix is

$\gamma^{0\oplus
2}_{NLO}=\left(\frac{\alpha_{s}}{4\pi}\right)^{2}\left(\begin{smallmatrix}\frac{97087}{972}-\frac{64f}{9}&\frac{6737}{324}&\frac{2f}{9}-\frac{1205}{108}&\frac{1717}{36}-\frac{2f}{3}&0&0&\frac{142q}{9}&\frac{2q}{3}\\\
\frac{281}{6}-\frac{1970f}{243}&\frac{818f}{81}+\frac{161}{6}&16-\frac{20f}{27}&\frac{202}{3}-\frac{4f}{3}&0&0&\frac{92q}{27}&\frac{52q}{9}\\\
-\frac{20525}{972}&\frac{19373}{324}&\frac{28f}{3}+\frac{11863}{108}&\frac{313}{36}-\frac{4f}{3}&0&0&-\frac{4q}{9}&\frac{4q}{3}\\\
-\frac{16f+18}{27}&\frac{208}{3}&2f+\frac{127}{2}&\frac{149-4f}{6}&0&0&-\frac{1664q}{243}&\frac{1232q}{81}\\\
-\frac{2q}{3}&2q&16q&0&\frac{553}{6}-\frac{58f}{9}&\frac{95}{2}-2f&-\frac{836}{243}&\frac{1700}{81}\\\
-\frac{1628q}{243}&\frac{1340q}{81}&\frac{88q}{27}&\frac{40q}{9}&\frac{95}{2}-2f&\frac{553}{6}-\frac{58f}{9}&\frac{46}{3}&2\\\
0&0&0&0&0&0&\frac{80f}{9}+\frac{43121}{486}&\frac{11095}{162}\\\
0&0&0&0&0&0&\frac{377}{6}-\frac{1322f}{243}&\frac{1178f}{81}+\frac{565}{6}\\\
\end{smallmatrix}\right)\,.$ (20)

We would like to emphasize that we cannot reduce the isospin structure of our
effective Hamiltonian further, in that we cannot separate it into $I=0$ and
$I=2$ pieces without further approximation. We note that the unique $I=2$
flavor structure of the operator shown in [27] and employed in [30] comes from
combining operators associated with $Z$ and $W$ exchange, and we cannot build
a pure $I=2$ operator unless we set the $\theta_{c}$, the Cabibbo angle, to
zero. This says, in effect, that the existence of more than one generation of
quarks precludes the possibility of writing the effective Hamiltonian into
components of purely definite $I$. Finally, as discussed in detail in [1], the
effects of charged-current interactions beyond tree level are numerically
small even in LO, so that we have included those effects as in [1].

##  4 RG evolution

Using the operators of Eq. (10), the effective Hamiltonian for HPV is [1]

$H^{\rm HPV\,[Z^{0}]}_{\rm
eff}(\mu)=\frac{G_{F}s_{W}^{2}}{3\sqrt{2}}\sum_{i=1}^{8}C_{i}(\mu)\Theta_{i}\,.$
(21)

We follow [23] in evolving the WCs, $C_{i}(\mu)$, from the $W$ scale,
$\mu_{W}$, to the $2\,\rm GeV$ scale, using the RG formalism at NLO. At NLO,
there are two separate aspects to consider. First, like in LO, the WCs at
energy scales below the $W$ scale, with $f=5$, are determined by

$\vec{C}(\mu)=U_{5}(\mu,\mu_{W})\vec{C}(\mu_{W})\,,$ (22)

though the evolution matrix $U(\mu,\mu_{W})$ at NLO takes the form

$U_{f}(\mu,\mu_{W})=\left(1+\frac{\alpha_{s}(\mu)}{4\pi}J\right)U_{f}^{(0)}(\mu,\mu_{W})\left(1-\frac{\alpha_{s}(\mu_{W})}{4\pi}J\right)\,,$
(23)

where $U_{f}^{(0)}(\mu,\mu_{W})$ is its LO form [1] with the correction matrix
$J$ given by

$J=VHV^{-1}\,.$ (24)

The matrix $V$ is comprised of eigenvectors that diagonalize the anomalous
dimension matrix, namely $\gamma_{D}^{(0)}=V^{-1}\gamma^{(0)T}V$. If we define
$G=V^{-1}\gamma^{(1)T}V$, then, the elements of the matrix $H$ are given by

$H_{ij}=\delta_{ij}\gamma_{i}^{(0)}\frac{\beta_{1}}{2\beta_{0}^{2}}-\frac{G_{ij}}{2\beta_{0}+\gamma_{i}^{(0)}-\gamma_{j}^{(0)}}\,,$
(25)

where $\gamma_{i}^{(0)}$ is a diagonal element of $\gamma_{D}^{(0)}$,
$\beta_{0}=(33-2f)/3$, and $\beta_{1}=102-38f/3$ [23]. The second aspect,
unique to NLO, comes from matching across a flavor threshold $f\to f-1$ at a
particular energy scale $m$. That is,

$\vec{C}_{f-1}(m)=M(m,f)\vec{C}_{f}(m)\,,$ (26)

where

$M(m,f)=1+\frac{\alpha_{s}(m)}{4\pi}\delta r_{(f)}^{T}$ (27)

and $\delta r_{(f)}=r_{(f)}-r_{(f-1)}$, with $\delta r_{(f)}$ characterized by
$\mathcal{O}(\alpha_{s})$ radiative corrections to the operators [23], which
come, as we have shown, purely from type-1 penguins. Finally, we have that

$\vec{C}(2\,{\rm GeV})=U_{4}(2\,{\rm
GeV},\mu_{b})M(\mu_{b},5)U_{5}(\mu_{b},\mu_{W})\vec{C}(\mu_{W})\,.$ (28)

In our LO analysis we studied the variation in the WCs for $\mu\in(1\,{\rm
GeV},4\,{\rm GeV})$ [1]. In NLO, we find the evolution matrix below charm
threshold becomes ill-defined due to the particular structure of the isovector
$\gamma^{(0)}$ matrix [1, 27]. Thus the lowest energy effective Hamiltonian we
compute is in a $2+1+1$ flavor theory at a scale of $1.5\,\rm GeV$.

###  4.1 Isosector RG evolution

For the prototype operators of Eq.(8), based on the results of [23] for
$\delta r_{(f)}^{T}$, the $r_{p1}$-matrix is

$r_{p1}(f)=f\left(\begin{smallmatrix}0&0&0&0\\\
\frac{10}{27}&-\frac{10}{9}&0&0\\\ 0&0&0&0\\\ 0&0&0&0\\\
\end{smallmatrix}\right)\,.$ (29)

Using this we can construct the $r_{p1}$ matrix pertinent to HPV, following
the procedure developed for the HPV penguin corrections. We report this matrix
for all isosectors, as well as for odd and even isosectors

$r_{p1}^{I=0\oplus 1\oplus 2}=\left(\begin{smallmatrix}0&0&0&0&0&0&0&0\\\
\frac{10f}{27}&-\frac{10f}{9}&0&0&0&0&0&0\\\ 0&0&0&0&0&0&0&0\\\
0&0&0&0&0&0&\frac{-10q}{27}&\frac{10q}{9}\\\ 0&0&0&0&0&0&0&0\\\
\frac{-10q}{27}&\frac{10q}{9}&0&0&0&0&0&0\\\ 0&0&0&0&0&0&0&0\\\
0&0&0&0&0&0&\frac{10f}{27}&-\frac{10f}{9}\end{smallmatrix}\right)$ (30)

$\\!\\!\\!\\!\\!r_{p1}^{I=1}=\left(\begin{smallmatrix}0&0&0&0&0&0&0&0\\\
\frac{10f}{27}&-\frac{10f}{9}&0&0&0&0&0&0\\\ 0&0&0&0&0&0&0&0\\\
0&0&0&0&0&0&0&0\\\ 0&0&0&0&0&0&0&0\\\
\frac{10q}{27}&-\frac{10q}{9}&0&0&0&0&0&0\\\ 0&0&0&0&0&0&0&0\\\
0&0&0&0&0&0&0&0\\\ \end{smallmatrix}\right)$

$\\!\\!\\!\\!\\!r_{p1}^{I=0\oplus
2}=\left(\begin{smallmatrix}0&0&0&0&0&0&0&0\\\
\frac{10f}{27}&-\frac{10f}{9}&0&0&0&0&0&0\\\ 0&0&0&0&0&0&0&0\\\
0&0&0&0&0&0&\frac{10q}{27}&-\frac{10q}{9}\\\ 0&0&0&0&0&0&0&0\\\
\frac{10q}{27}&-\frac{10q}{9}&0&0&0&0&0&0\\\ 0&0&0&0&0&0&0&0\\\
0&0&0&0&0&0&\frac{10f}{27}&-\frac{10f}{9}\end{smallmatrix}\right)$ (31)

for subsequent use in our RG analysis. Using the inputs of [1], the strong
interaction strength ratios at NLO are calculated employing the RunDec package
[32] to yield: $\alpha_{s}(M_{b})/\alpha_{s}(M_{W})=1.86$ and
$\alpha_{s}(2\,{\rm GeV})/\alpha_{s}(M_{b})=1.35$. Finally, we are in a
position to compute the RG evolution from $M_{W}$ to $2\,\rm GeV$, finding

$\vec{C}^{I=1}=\left(\begin{smallmatrix}\underline{M_{W}}\\\ 1\\\ 0\\\ 0\\\
0\\\ 3.49\\\ 0\\\ 3.49\\\
0\end{smallmatrix}\right)\longrightarrow\left(\begin{smallmatrix}\underline{2\,{\rm
GeV}}&\underline{[1.5\,{\rm GeV}-4\,{\rm GeV}]}\\\ 0.955&[0.947\dots 0.964]\\\
-0.009&[-0.007\dots-0.007]\\\ 0.107&[0.139\dots 0.053]\\\
-0.377&[-0.446\dots-0.253]\\\ 3.83&[3.98\dots 3.72]\\\
-1.28&[-1.49\dots-0.906]\\\ 3.83&[3.89\dots 3.72]\\\
-1.28&[-1.49\dots-0.906]\end{smallmatrix}\right)\;;$ (32) $\vec{C}^{I=0\oplus
2}=\left(\begin{smallmatrix}\underline{M_{W}}\\\ -1\\\ 0\\\ 0\\\ 0\\\ -3.49\\\
0\\\ 0\\\
0\end{smallmatrix}\right)\longrightarrow\left(\begin{smallmatrix}\underline{2\,{\rm
GeV}}&\underline{[1.5\,{\rm GeV}-4\,{\rm GeV}]}\\\
-0.976&[-0.966\dots-0.978]\\\ -0.029&[0.011\dots 0.007]\\\
-0.091&[-0.192\dots-0.106]\\\ 0.446&[0.488\dots 0.286]\\\
-3.83&[-3.97\dots-3.75]\\\ 1.28&[1.36\dots 0.864]\\\
-0.129&[-0.080\dots-0.050]\\\ 0.303&[0.438\dots
0.226]\end{smallmatrix}\right)\,.$ (33)

With this we have determined the isospin-separated form of Eq. (21, noting
Eqs. (17,19) as used in [1], to find the effect of $Z^{0}$ exchange at $2\,\rm
GeV$. Combining this with our earlier analysis of the tree-level charged-
current contributions [1], we can write the pertinent effective PV Hamiltonian
as

$\mathcal{H}_{\rm
eff}^{I}(\mu)=\frac{G_{F}s_{w}^{2}}{3\sqrt{2}}\sum_{i=1}^{12}C_{i}(\mu)^{I}\Theta_{i}^{I}(\mu)\,.$
(34)

The WCs and the local operators here depend on both the energy scale $\mu$ and
the renormalization scheme used in calculating the NLO QCD corrections.
However, physical amplitudes computed using our effective Hamiltonian are
proportional to the product of $C_{i}(\mu)$ and hadronic matrix elements
$\braket{\Theta_{i}(\mu)}$, yet they should be independent of these purely
calculational effects. That is, the scheme and scale dependence of one should
cancel that of the other, up to still higher-order effects [23].
Unfortunately, as yet, fully non-perturbative evaluations of
$\braket{\Theta_{i}(\mu)}$ are not yet available, and as a result our
determinations will be scale and scheme dependent. We will comment further on
this in the context of an explicit example.

###  4.2 Application to meson-nucleon parity-violating couplings

We can use our NLO-improved effective Hamiltonian to compute parity-violating
meson-nucleon couplings, thus updating our LO analysis in [1, 2]. In all this
work, we use the factorization approximation in the evaluation of the hadronic
matrix element of the four-quark operators, as well as precise lattice QCD
(LQCD) determinations of the quark-flavor charges of the nucleon [33, 34], as
in [1, 2]. These outcomes can be compared to constraints on these quantities
deduced from low-energy experiments analyzed within a hadron-based framework
[19, 9, 11]. We thus come to our NLO updated results at $\mu=2\,\rm GeV$,
noting our earlier LO results in brackets throughout. That is,

$\begin{split}h_{\pi}^{1}&=2.14\pm
0.21+\left(\stackrel{{\scriptstyle{+0.17}}}{{{}_{-0.34}}}\right)\times
10^{-7}\quad[3.06\pm
0.34+\left(\stackrel{{\scriptstyle{+1.29}}}{{{}_{-0.64}}}\right)\times
10^{-7}]\,,\end{split}$ (35)

to be compared with the experimental determination $h_{\pi}^{1}=(2.6\pm
1.2_{\rm stat}\pm 0.2_{\rm sys})\times 10^{-7}$ [17], while noting that the
bound from 18F radiative decay is $|h_{\pi}^{1}|<1.3\times 10^{-7}$ at 68% CL
[16]. Our error estimates come from the LQCD inputs and from the change in WC
over a scale variation of $1.5\,\rm GeV$ ($1.0\,\rm GeV$ in the LO case) to
$4\,\rm GeV$ (lower entry), respectively. Moreover,

$\displaystyle h^{1}_{\rho}=-0.275\pm
0.040+\left(\stackrel{{\scriptstyle{+0.006}}}{{{}_{-0.002}}}\right)\times
10^{-7}\quad$ $\displaystyle[-0.294\pm
0.045+\left(\stackrel{{\scriptstyle{0.014}}}{{{}_{-0.036}}}\right)\times
10^{-7}]$ $\displaystyle h^{1}_{\omega}=1.58\pm
0.10+\left(\stackrel{{\scriptstyle{+0.01}}}{{{}_{-0.02}}}\right)\times
10^{-7}\quad$ $\displaystyle[1.83\pm
0.11+\left(\stackrel{{\scriptstyle{-0.05}}}{{{}_{0.13}}}\right)\times
10^{-7}]$ $\displaystyle h^{0}_{\omega}=0.277\pm
0.014+\small{\left(\stackrel{{\scriptstyle{+0.008}}}{{{}_{-0.34}}}\right)}\times
10^{-7}\quad$ $\displaystyle[0.270\pm
0.015+\left(\stackrel{{\scriptstyle{-0.32}}}{{{}_{0.55}}}\right)\times
10^{-7}]\,$ $\displaystyle h_{\rho}^{0}=-10.6\pm
0.6+\small{\left(\stackrel{{\scriptstyle{+0.02}}}{{{}_{+0.9}}}\right)}\times
10^{-7}\quad$ $\displaystyle[-11.1\pm
0.7+\left(\stackrel{{\scriptstyle{1.1}}}{{{}_{-2.1}}}\right)\times 10^{-7}]\,$
$\displaystyle h_{\rho}^{2}=9.27\pm
0.67+\left(\stackrel{{\scriptstyle{-0.41}}}{{{}_{+0.85}}}\right)\times
10^{-7}\quad$ $\displaystyle[8.57\pm
0.52+\left(\stackrel{{\scriptstyle{1.12}}}{{{}_{-1.74}}}\right)\times
10^{-7}]\,.$

Computing the empirically determined combination $h_{\rho-\omega}\equiv
h_{\rho}^{0}+0.605h_{\omega}^{0}-0.605h_{\rho}^{1}-1.316h_{\omega}^{1}+0.026h_{\rho}^{2}=(-17.0\pm
6.56)\times 10^{-7}$ [18], we have

$\begin{split}h_{\rho-\omega}&=-12.2\pm
0.62+\left(\stackrel{{\scriptstyle{0.07}}}{{{}_{0.74}}}\right)\times
10^{-7}\quad[-12.9\pm
0.52+\left(\stackrel{{\scriptstyle{0.97}}}{{{}_{-1.9}}}\right)\times
10^{-7}]\,.\end{split}$ (37)

We observe that both our $h_{\pi}^{1}$ and $h_{\rho-\omega}$ are within $\pm
1\sigma$ of their experimental determinations, and our NLO values tend to be
smaller than our LO results. The WCs are scale dependent, but the scale
sensitivity becomes much reduced in moving from LO to NLO. Thus our NLO
results show less variation about the $\mu=2\,\rm GeV$ scale.

##  5 Summary

We have presented a method of determining the anomalous dimension matrix
apropos to parity-violating, $\Delta S=0$ hadronic processes at NLO in QCD. We
have also used its outcomes to employ a NLO RG analysis to compute the
associated parity-violating weak effective Hamiltonian for $2+1+1$ flavors at
a renormalization scale of $2\,\rm GeV$, starting from the full SM at
$\mu=M_{W}$. Consequently, operator mixing at NLO, as well as its modification
across heavy-flavor thresholds, has been calculated for each possible isospin
sector ($I=0,1,2$) for HPV. At the energy scales of interest to us, the
effects of quark masses and of external momenta are negligible, making the
gluon interactions that appear in radiative corrections to HPV operators
flavor blind. With this, the crucial idea is that we can logically deduce the
anomalous dimension matrices for the operators under study from similar
computations of other processes, provided enough information is available to
construct the mixing profile of a set of prototype four-quark operators. Here
we have done this using the computations of anomalous dimension matrices in
$|\Delta F|=1$ processes in NLO QCD [22, 23], by exploiting the parity
symmetry of the radiative corrections in QCD to build prototype operators that
bridge from flavor-changing operators to parity-violating, flavor-conserving
ones. As a test of our procedures, the anomalous dimension results in LO have
been reproduced, too, but are not presented here. With these results in place,
we have refined our earlier studies of parity-violating meson-nucleon
couplings, to find better agreement with low-energy experiments.

## Acknowledgments

We acknowledge partial support from the U.S. Department of Energy Office of
Nuclear Physics under contract DE-FG02-96ER40989.

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